Math Power Guide

20092010E D I T I O N

MATHALGEBRATHROUGHCALCULUS

MATHPOWER GUIDEWRITERS

Julia Ma & Steven Zhu CONTRIBUTIONS& REVISIONS

Michael Nagel

EDITORS

Dean Schaffer & Sophy Lee ALPACA-IN-CHIEF

Daniel Berdichevsky

®

the World Scholar’s Cup®

DOING OUR BEST, SO YOU CAN DO YOURS

DemiDec, The World Scholar’s Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation. Academic Decathlon and USAD are registered trademarks of the United States Academic Decathlon Association.

DemiDec is not affiliated with the United States Academic Decathlon.

MATH POWER GUIDE®

I. WHAT IS A POWER GUIDE?........................................................ 2 II. CURRICULUM OVERVIEW............................................................3 III. GENERAL MATH..............................................................................4 IV. ALGEBRA…........................................................................................ 10 V. GEOMETRY………….......................................................................... 40 VI. TRIGONOMETRY.............................................................................49 VII. CALCULUS……………………………………………………………………..... 56 VIII. POWER LISTS................................................................................... 66 IX. POWER TABLE..................................................................................73 X. POWER STRATEGIES…………………………………………………………74 XI. ABOUT THE AUTHORS..................................................................76

BY

JULIA MA

CALIF. INSTITUTE OF TECHNOLOGY

ALTA HIGH SCHOOL

STEVEN ZHU

HARVARD UNIVERSITY

FRISCO HIGH SCHOOL

EDITED BY

DEAN SCHAFFER

STANFORD UNIVERSITY

TAFT HIGH SCHOOL

SOPHY LEE HARVARD UNIVERSITY

PEARLAND HIGH SCHOOL

DEDICATED TO ALPACAS

© 2009 DEMIDEC

Math Power Guide | 2

DemiDec, The World Scholar’s Cup, Power Guide, and Cram Kit are registered trademarks of the DemiDec Corporation. Academic Decathlon and USAD are registered trademarks of the United States Academic Decathlon Association.

DemiDec is not affiliated with the United States Academic Decathlon.

WHAT IS A POWER GUIDE? “ .”

That’s the sound of 10,000 points looming silently at your doorstep. Although they don’t have weapons, they do arrive armed with ideas—vague but powerful ones, like “Liberty,” “Equality,” and “Fraternity.”

You can net 3,000 of these points through Speech, Interview, and Essay. That leaves 7,000 that you’ll have to earn, question by question, test by test.

Do you run? No! Do you hide? No! Do you catch ‘em all? Yes!

After all, you have on your side 10 formidable Power Guides. Each is stocked with every single testable fact that you will need to know this year. DemiDec Resources will teach you the material. Workbooks will drill the information. Power Guides will make sure that not a single nuance from the curriculum falls through the cracks. At the very end of the Power Guide, you will find a collection of Timelines, Power Tables, and glossary-like Power Lists to help you sweep up every point possible.

Math. The very word strikes fear into the hearts of many. But don’t be discouraged—math, like any other event, can be mastered through studying, and perhaps more than any other event, through test-taking. Sounds simple, right?

Unfortunately, Decathlon math is so broad that no guide could possibly hope to cover all of nooks and crannies. This Power Guide, then, is meant as a quick review tool, a cram kit writ large, not a learning tool. I advise you to go through this guide with textbooks nearby. I’ve found time and time again that doing example problems is the best way to reinforce the concepts that you learn.

So what are you waiting for? It’s now or never: pick up your calculators, sharpen your pencils, and rebel—er—review!

Sincerely,

Sophy Lee


CURRICULUM OVERVIEW The breakdown of exam questions will be as follows: general math, 10%; algebra, 30%; geometry, 30%; trigonometry, 20%; and differential calculus, 10%. This is shown in the pie chart below.

Calculus questions tend to be simple (and, thus, mastered with limited studying), but not many of them appear on the test. The same goes with the general math portion. Algebra and geometry will undoubtedly form the core of the test, so knowing these areas is key to scoring well.

For math, it’s especially important to remember that many concepts will appear in other sections. While “general math” is technically only supposed to be tested in five questions, general math concepts can (and definitely will) appear in problems from the other categories, such as algebra and trig.

Differential Calculus

10%

Algebra30%

Geometry30%

Trigonometry20%

General Math10%


GENERAL MATH

Integers, Fractions, Percents, Decimals Integers

An integer is a whole number, including 0 -1, 4, and 934 are integers -2/3, 4/5, and 0.2 are not integers

Fraction arithmetic To add and subtract fractions, find a common denominator

Multiply each fraction by a fraction form of n/n to make all the denominators equal

Example: 2113=

216+

217=

33×

72+

77×

31=

72+

31

Remember to abide by the order of operations: parentheses, exponents, multiplication and division, addition and subtraction To remember the order of operations, try this mnemonic: Please Excuse My

Dear Aunt Sally Multiplying fractions is done “straight across”

Numerators are multiplied together, and denominators are multiplied together

Example: 212=

7×32×1=

72×

31

To divide fractions, multiply by the reciprocal of the dividend The “dividend” is the second fraction (the one that is being divided into the first) To find the reciprocal of a fraction, simply flip it over

Example: 67=

27×

31=

72÷

31

Commonly, fractions come into play when problems describe how fast two people work and the time it takes for them to finish a job if they work together Example: Joe can paint a fence in 3 hours, and Sally can paint 2 fences in 5 hours; how

long will it take them to paint four fences together?

Joe’s rate of work is 1 fence every 3 hours or 31 fence per hour

Sally’s rate of work is 2 fences every 5 hours or 52 fence per hour

POWER PREVIEW POWER NOTES

The topics covered in general math are relevant in our daily lives, though we often don’t realize it. Whether you are figuring out how much to tip a waiter or trying to decide on matching socks, math lurks in the corner of countless activities.

According to the USAD outline, 3-4 questions (10% of the test) will come from this section

General math is not covered in the USAD math basic guide


The formula for work is work = rate× time Since we’re trying to find time, let’s divide rate from both sides

time=rate

work

To answer the problem, the set-up is t=

hours5fences2

+hours3fence1

fences4

We’re dividing the total work (4 fences) by the total rate of work (the combined speeds of Joe and Sally) to find the time t that it takes for them to finish

hours45.5=11hours60

=fences11hours15

×fences4=

hours15fences11

fences4=t 1

Percentages

1% represents 100

1 of the whole

Converting a percentage to a decimal involves dividing by 100

Example: 12% equals 12.0=10012

Converting a decimal to a percentage involves multiplying by 100 Example: 0.73 equals %73=%100×73.0

Percentages are generally applied to problems dealing with sales (discounts and sales taxes) A typical problem asks about an item with a discount of x%

The discounted price is equal to )priceoriginal)(100

x-1(

Example: A $20 shirt with a 30% discount sells for (1 – .30)(20) = (.70)(20) = $14

The total cost of an item with a tax of x% equals )priceoriginal)(100

x+1(

Example: A $10 hat with 6% tax is sold at (1.06)($10) = $10.60

The discount received, given the original and selling prices, equals price originalprice selling-1

Example: Someone who buys a watch at $20 when the original price was $25 receives

a discount of %20=20.0=2520-1

The order of multiple discounts does not affect the end price Example: A $20 book on sale at 20% off that is bought with a coupon for an

additional 25% discount will cost the same as a $20 book on sale at 25% off that is bought with a coupon for an additional 20% discount Both would cost ($20)(0.80)(0.75) = $12

The original price will not affect the percentage discount

1 A note on rounding: remember that rounding depends on the place to which you are asked to round. Look only at the digit after that one. Were we to round this number to an integer, it would be 5 because there is a 4 in the tenths place. Somewhere, some poor elementary math teacher is confusing students by telling them that the 5 in the hundredths place would make the 4 in the tenths place round up to 5, which would then cause the 5 in the ones place to round up to 6. This is NOT how to round, as 0.45 is obviously less than half and could only round down the ones place.


Example: Any item at 30% off with a sales tax of 6% will always be (0.70) (1.06) = 0.742 = 74.2% of the original price

Another typical problem asks about the original price of an item before a discount Example: What is the price of a shirt that costs $10.32 after a 20% discount and

7.5% tax?

Here’s the set-up: 12$=0.20-1

)075.0+132.10$(

=p

Since tax is included, we had to divide it from the final price The order of division, however, does not matter, just as with multiple discounts

wellas12$=0.075+1

)20.0-132.10$(

=p

ATTENTION: Excise taxes (taxes on specific items) do not compound other taxes Excise taxes are based on the item’s original price Example: A meal has an original price of $10, an 8% sales tax, a 10% tip, and a 4%

excise tax; what is its final price? With the sales tax and tip, the cost is $10× (1 + 0.08) × (1 + 0.10) = $11.88 The excise tax is calculated from the original price: $10× 0.04 = $0.40 The final cost is $11.88 + $0.40 = $12.28

Counting The multiplication principle

The multiplication principle helps us find the total number of possibilities when we are choosing one item from each of several groups Multiply the total number of items in each group to find the total number of possibilities

Example: There are 3 kinds of computers, 4 kinds of monitors, and 2 types of mice The total number of ways to pick a unique combination of each is 3× 4× 2 = 24

Permutations We use permutations to find the total number of possible arrangements of a given set of

objects Order is important

Example: ABCD is a different arrangement from DBAC To find the total number of possible arrangements for r objects out of n total objects,

calculate r)!-n(!n=Prn

n! = (n)(n-1)(n-2)…(2)(1) n! is called a factorial2 Take the number n and keep multiplying by the integers between n and 0 9! would be 9× 8× 7× 6× 5× 4× 3× 2× 1 = 362,880

Example: There are 5 runners in a race; how many different possibilities are there for the top three places?

2 The exclamation point is, sadly, not for emphasis. N!!! – Steven


Order is important, so do a permutation

60=1×2

1×2×3×4×5=3)!-5(!5=P35

Combinations We use combinations to find the total number of possible arrangements of a given set of

objects when order does not matter3 Example: A committee of John, Jake, and Joe is the same as a committee of Joe, John,

and Jake To find the total number of combinations for r objects out of n total objects, calculate

For the same r and n, there will be fewer combinations than permutations by a factor of

r!

!rP=C rn

rn

Example: There are 10 students in Mr. Jacob’s class, and three will get to serve on the student council; in how many ways can the students be selected for the council? Order does not matter, so do a combination

120=)1×2×3×4×5×6×7)(1×2×3()1×2×3×4×5×6×7×8×9×10(=

)!7)(!3(!10=C310

The arrangement principle The arrangement principle allows us to calculate the total number of possible arrangements

when some of the items we are examining are identical To arrange n objects where r objects are indistinguishable, divide n! by r!

Each set of identical objects should be considered as a separate r!

Example: The number of arrangements for the letters in the word “choose” is 360=!2!6

There are 6 letters with 2 non-distinct o’s Example: The number of arrangements for the letters in the word “Mississippi” is

650,34=)!2)(!4)(!4(

!11

There are 11 letters total, with 4 s’s, 4 i’s, and 2 p’s Arranging objects in circles

Test questions occasionally ask the total number of ways to arrange items around a circle For example, a question might ask how many different ways 7 people can sit around a

circular table When we arrange objects in circles, we need to make sure that our arrangements are actually

different in order rather than just rotated clockwise or counter-clockwise

3 The word “combination” might get you thinking about your locker combination. Interestingly enough, locker combinations are actually permutations: the order in which you enter the numbers does matter. – Dean


In the above diagram, circle B has the same order as circle A, just rotated clockwise

Starting with 1 and moving clockwise, their orders are both 1, 2, 3, 4 The figures cannot be distinguished from each other by order, so we must not count

both of them when finding the total number of arrangements Figure C, however, has a different order

Starting with 1 and moving clockwise, its order is 1, 4, 3, 2 To make sure that we only count different orders in circular arrangements, we will have

one object stay in the same place so we can tell that the other objects have moved around Since we’re keeping one object in place for circles, we will use (n – 1)! to find the total

number of arrangements that have different orders When we arrange objects in lines, we use n! to find the total number of arrangements Example: How many different ways can five people sit around a circular table?

We keep one person in place and let the other four move around (5 – 1)! = 4! = 24! different arrangements

Example: How many different ways can five keys be arranged on a circular keychain? Like the table problem, we keep one key in place and move the other four around Unlike the table problem, we also need to divide the arrangements in half because half of

the arrangements are repeated if we flip the keychain over4

12=224=

2!4=

21)!-5( different arrangements

Calculator strategy Most scientific and graphing calculators have factorials, combinations, and permutations as

programmed functions Being familiar with the keys to access these functions will save you time on the test if you can

identify which ones to use for each problem

4 Watch for similar scenarios such as beads on a bracelet or necklace (both of which can be flipped over).

1

2

3

4

4

1

2

3

1

4

3

2 A B C


Probabilities Introduction

Probability is the chance that a certain event will happen The probability of event A occurring is defined as P(A), where

outcomesofnumberpossibletotaloutcomestotalofnumber

=)A(P

Note that this formula only works when all outcomes are equally likely Example: A player rolls a standard die, wanting a number greater than 3

Event A is “getting a number greater than 3” (rolling a 4, 5, or 6)

The probability this will happen is 5.0=21=

63=)A(P

More examples Use the counting principles to help calculate probability

Example: A club of 10 people wants to select a president If the president is randomly selected, then the probability of a specific person being

selected is 101=

C1=)A(P

110

Example: When calculating poker probabilities for a 52-card deck and a 5-card hand, the denominator will always be the total number of hands possible, in this case 52 5C The 52 on the lower-left hand corner of the C stands for the total number of items

from which you can choose (52 cards in the entire deck) The C denotes “combination” The 5 on the lower-right hand corner of the C stands for the total number of items

that we are choosing (5 cards in a hand) Often, problems ask about the probability of rolling a certain sum with two dice

In these cases, the denominator will always be 36=6×6 To find the numerator, the easiest way is to list the outcomes for the desired sum Example: Find the probability of rolling a sum of 7 with two dice

The possible outcomes for a sum of 7 are 1-6, 6-1, 2-5, 5-2, 3-4, and 4-3

So, with six total outcomes, the probability is 61=

366


ALGEBRA

Polynomial Equations Introduction

An equation is a mathematical statement that two expressions are equal Example: 4 + 1 = 5 Example: 2x + 2 = 10

Linear and quadratic equations A linear equation is an equation in which the highest power of the variables is 1

x=y is a linear equation The graph is a straight line

y = x2 is not a linear equation because x has a power of 2 The graph is a parabola

Often, a problem will ask you to solve for a variable To do this, isolate the variable in question by performing equivalent operations on

both sides of the equation Example: Solve y = mx + b for m for m

Subtract b from both sides: y – b = mx

Then divide both sides by x to get x

b-y=m

A linear equation is graphed as a straight line The slope-intercept formula of a linear equation is y = mx + b

m is the slope of the line

m = 21

21

x-xy-y

for points (x1, y1) and (x2, y2)

Vertical lines have no slope (or infinite slope) Another way to put it is that the slope is “undefined” A vertical line looks like an “I” (for “Infinite slope”) A vertical line is also the first stroke of “N” (for “No slope”)

Horizontal lines have 0 slope A horizontal line is the first stroke of “Z” (for “Zero slope”)

Do not confuse 0 slope with “no slope” “No slope” means that the slope is nonexistent Zero slope has a value, which is 0

5 Hopefully, surgical treatment won’t be necessary when you’re done with this section. – Dean


Algebra was brought from ancient Babylon, Egypt, and India to Europe via the Arabs. The term derives from the Arabic al-jabr or, literally, “the reunion of broken parts.” In addition to its mathematical meaning, the word also refers to the surgical treatment of fractures.5


Covers pages 4-25 in the USAD math basic guide


(0, b) is the y-intercept of the line The y-intercept is the point where the line intercepts the y-axis At the y-axis, x = 0

Example: Find the equation of the line that passes through (-4, 7) and (3, -2) As the name of the formula implies, you must find the slope first

7-9=

3-4-(-2)-7=m

Then we move on to the intercept We have m, so to find b, we need to plug in a point for x and y You can choose either of the points given, but we’ll use the second one

because it has smaller numbers

Our equation looks like this: b+)3(79- = 2-

713=b

Now that we have the slope and the intercept, we can write the equation of the line

713+x

79-=y

The point-slope formula of a linear equation is y – y1 = m(x – x1) In this equation, (x1, y1) is any point on the given line Just as in slope-intercept form, m is the slope of the line Example: What is the equation of the perpendicular bisector of a line segment with

endpoints (2, -2) and (4, 4)? First, we need to find the slope of the line segment

m = 3=2-6-=

4-24-2-

The slope of any perpendicular line will always be the negative reciprocal of the original slope

m = 3-1

The bisector will pass through the midpoint of the line segment, so we need to find those coordinates Remember that the coordinates of the midpoint are the averages of the

endpoints

At the midpoint, x = 3=2

4+2 and y = 1=2

4+2-

The midpoint is at (3, 1) Now, we can plug the point and the slope into the formula

y – 1 = )3-x(31-

In slope-intercept form, the answer is y = 2+x31-

The standard form of a line is written as ax + by = c


b-a is the slope and (0,

bc ) is the y-intercept

In the example above, the answer rewritten in standard form is x + 3y = 6 A quadratic equation is an equation in which the highest power of the variables is 2

Notice that if the product of two expressions is 0, then one or both of the expressions must also be 0

To solve an equation, use factoring: use the distributive property backwards If the equation’s form is Ax2 + Bx + C = 0, factor it out to (ax + b)(cx + d) = 0

ac = A, bd = C, and (ad + bc) = B If the form of the equation is x2 + Bx + C = 0, factor it out to (x + c)(x + d) = 0

cd=C and (c + d)=B Once the equation is factored, each factor can be set to 0 to solve for x

Example: If (x + a)(x – b) = 0, then (x + a) = 0 or (x – b) = 0 This means x = -a or x = b

a and b are each called roots of the quadratic equation They are also called zeros (because the equation equals zero when they are

plugged in for x) and x-intercepts The quadratic formula6 can also be used to find roots

Given ax2 + bx + c = 0, x = 2a

(4ac)-b±b- 2

Example: Solve for the roots of 12x2 – 7x – 10 = 0 We can factor the equation into (4x – 5) and (3x + 2)

Then, we set each factor equal to 0 to find the roots

4x – 5 = 0, x = 45

3x + 2 = 0, x = 32-

We can also use the quadratic formula to get the same answer

45=

2(12)4(12)(-10)-(-7)+(-7)-=x

2

32-=

2(12)4(12)(-10)-(-7)-(-7)-=x

2

FOIL To convert a factored quadratic back to ax2 + bx + c form, use the FOIL process

FOIL stands for first, outer, inner, last Start by multiplying the first parts of each term Next, multiply the “outside” parts of the factored form Then, multiply the “inside” parts of the factored form Next, multiple the last parts of each term Finally, take the sum of the products and combine like terms

Example: convert (3x + 7)(x + 5) to ax2 + bx + c form “First”: (3x)(x) = 3x2

6 Many algebra students become familiar with this formula by singing it to the tune of “Pop Goes the Weasel.” It goes: x equals negative b/ plus or minus the square root/ of b squared minus 4ac/ all over 2a. – Steven


“Outer”: (3x)(5) = 15x “Inner”: (7)(x) = 7x “Last”: (7)(5) = 35 3x2 + 15x + 7x + 35 = 3x2 + 22x + 35

Higher order equations Higher order equations are equations in which the highest power of the variables is greater

than 2 Sometimes, with a suitable substitution, we can solve a higher order equation like a quadratic

equation When we looked at quadratic equations, the first term had a power of 2, and the second

term had a power of 1 Likewise, we can solve higher order equations if the power of the first term is double the

power of the second term Example: Find the roots of x6 + 2x3 + 1 = 0

The power of the first term is 6, and the power of the second term is 3 To change the equation into a quadratic equation, we want the second term to have a

power of 1 and the first term to have a power of 2 Thus, we will substitute u for 3x The equation turns into u2 + 2u + 1 = 0 Factoring gives us (u + 1)2 = 0 u = –1 Remember, though, we are looking for x, not u Because u = –1 and u = x3, we know that x3 = –1 x = 3 1-

x = –1 The sum of cubes and differences of cubes formulas can also be used to solve some cubic

equations (equations in which the highest power of the variables is 3) Sum of cubes formula: (x3 + y3) = (x + y)(x2 – xy + y2) Difference of cubes formula: (x3 – y3) = (x – y)(x2 + xy + y2)

Factors and roots The remainder theorem and factor theorem are used to determine the remainders or factors,

respectively, of a polynomial Remainder theorem: if f(x) is a polynomial, then f(c) is the remainder when f(x) is

divided by (x – c) Example: Find the remainder when x6 – 3x5 + 7x4 – 2x3 – 12x2 + x – 5 is divided by

(x + 3) Here, c = –3 because the divisor7 is (x – (–3)) Now, we plug c into the dividend (–3)6 – 3(–3)5 + 7(–3)4 – 2(–3)3 – 12(–3)2 + (–3) – 5 = 1963

Make sure you use the parentheses when you punch this expression into your calculator

7 Remember, when division is written as a fraction, the dividend is on the top, and the divisor is on the bottom.


If your calculator allows you to store variables, storing –3 as a variable may help you avoid mistakes and speed up your typing because then you won’t need parentheses

The remainder is 1963 Factor theorem: if you use the remainder theorem, and the remainder equals 0, then (x –

c) is a factor of f(x) The rational roots theorem is used to determine all the possible rational roots of a

polynomial We apply the rational roots theorem to polynomials in the form of Axn + Bxn-1 +…C

In this form, A is the leading coefficient—the number in front of the term with the highest power—and C is the constant Both A and C must be integers

We must first find all the factors of A and all the factors of C We’ll use q to represent all the factors of C, and we’ll use p to represent all the factors

of A According to the theorem, all of the rational real roots can be found with the

expression pq±

To find all the possible roots, plug the various factors into the above expression Example: Find the possible rational roots of 36 + 2x3 + x4 – 11x2 – 12x

First, we make sure that we spot the correct coefficient for A The highest power is 4, and the coefficient for that term is 1

Thus, A = 1 The constant C is 36 Now, we list all the factors of C over all the factors of A

The possible rational roots are 136±,

118±,

112±,

19±,

16±,

14±,

13±,

12±,

11±

Luckily, A was 1 in this case Had A been 6, we would’ve had to list all the factors of C over 1, 2, 3, and 6,

resulting in four times as many possible rational roots After listing all the possible rational roots, we can use the factor theorem to find the

actual roots Sometimes you will need to find the sum or product of the roots, but not the roots

themselves

The formula to find the sum of the roots is ab- for all polynomials, where a is the

leading coefficient, and b is the coefficient of the second-highest degree term Example: Find the sum of the roots of 4x2 – 7x + 5

We will use ab-

a = 4 b = –7

The sum of the roots is –47=

4-7

Example: Find the sum of the roots of x3 + 3x2 – 4x – 12


We will use ab-

a = 1 b = 3

The sum of the roots is 3=13

The formula to find the product of the roots is ac- for odd-numbered polynomials

and ac for even-numbered polynomials, where a is the leading coefficient, and c is

the constant Example: Find the product of the roots of 5x2 + 8x – 2

a = 5 c = –2

The product of the roots is 52-

Solving Inequalities Inequality: a definition

An inequality states that two expressions are not equal Example: 4 + 5 < 12 Example: 4x + 2 > 3y – 4

Linear and quadratic To solve a linear inequality, treat the inequality as an equation and isolate the variable

Be careful to flip the sign if you multiply or divide by a negative number Example: –3x + 7 > 5

We subtract 7 from both sides to get the term with x by itself –3x > –2 Then we divide both sides by -3 and flip the sign

x < 32


In the above graph, the inequality is true in the shaded area, which is the region

to the left of, but not including, x = 32

The answer may also appear in the form of a number line

The open circle means that the value

32 is not included in the solution

If the inequality were x ≤ 32 , then the number line would look like this:

The darkened circle means that the solution includes

32

Linear inequalities can have more than one variable

Example: y ≤ 2+x31

To graph this inequality, we must plot the line and then shade the region above it

At the line, y is equal to the function

10-1 -2 -3 2 3

10-1 -2 -3 2 3


We shade above the line because y can also be greater than the function

Example: y < –2x + 1

Now, we shade below the line because y is less than the function

Usually in graphs, “less than” looks the same as “less than or equal to,” and

“greater than” looks the same as “greater than or equal to” Sometimes when y is not equal to the function, the line of the function is

dotted rather than solid But really, the shading is the important part


To solve a quadratic inequality, treat it like an equation and solve for the roots After finding the roots, place them on a number line The roots will partition the number line into different regions Test numbers in each region

The ones that make the inequality true will be part of the solution Example: x2 + 6x – 7 < 0

First, we factor (x + 7)(x – 1) < 0 Our roots are -7 and 1 We will place these roots on a number line

Notice that the roots divide the line into three regions: x < –7, –7 < x < 1, and x > 1 We will choose a number in each region to test the inequality Let’s use -8, 0 and 2

Rule of thumb8: whenever you can choose 0 as a test value, do so, as it is usually very easy to plug into the inequality

When we plug -8 into the inequality, we get 9 < 0, which is false When we plug 0 into the inequality, we get –7 < 0, which is true When we plug 2 into the inequality, we get 9 < 0, which is false On the number line, we will place an x where the inequality is false and a check

where the inequality is true

Thus, the solution to the quadratic inequality is –7 < x < 1

Quadratic inequalities may have an x and a y

8 If you’ve ever seen The Boondock Saints, the line at the beginning about “rule of thumb” is most excellent. – Steven

-7 1

-7 1


Example: y ≤ 2+x 2 We can represent this inequality graphically

\

Since y is less than or equal to the function, we shade the area below the curve, and the area includes the curve itself

Absolute value A number’s absolute value is essentially its distance from 0 on a number line

It is always non-negative Example: 45=45- Example: 45=0

When the expression inside the absolute value signs is a function, we set the function equal to two opposite values Example: 2=3 -x

x – 3 = 2 x – 3 = –2 x = 1 or x = 5

When we have inequalities with absolute values, we have to be careful with the direction of the inequality symbol Example: 6 +x ≥ 7

For the first inequality, we just remove the absolute value signs 6+x ≥ 7 For the second inequality, however, we have to flip the symbol of inequality because

we change the sign of the value to the right of the symbol 6+x ≥ –7

The value to the right of the inequality is now negative 7, and the symbol is now less than or equal to


Thus, x ≥ 1 or x ≤ 3 – 13 Example: 7 -2x < –3

Since an absolute value can never make an expression negative, the inequality can never be true

Watch out for trick questions like this one

Functions: Rational, Exponential, and Logarithmic Functions

For each possible value of a given independent variable, x, of a function, there can be only one value of the dependent variable, y If you plug in a value for x, you should get one value for y

Example: y = f(x) = x2 When x = 2, f(2) = (2)2 = 4 f(x) denotes that y is a function of x

More than one value of x may have the same value for y Example: y = f(x) = x2

f(2) = 4 f(–2) = 4 In this case, y was 4 when x was 2 and when x was -2

No value of x, however, may have more than one value for y Example: y = f(x) = x

f(9) = 9 = 3 f(9) = 9 = –3 In this case, y may not equal both 3 and -3

For an equation to be a function, it must pass the vertical line test A vertical line placed anywhere on the graph of a function can cross the function in at

most one point If the line intersects the graph at more than one point, then it isn’t a function

We can place a vertical line anywhere on the graph above, and it would only cross

the graph at one point, which means that the graph represents a function


A vertical line will cross the graph above at two places when x > 0

This graph, therefore, does NOT represent a function The domain of a function is all possible values of x (the independent variable)

Any value of x that causes a mathematical error in the function is NOT included in the domain

Possible limitations on domain include dividing by 0, taking the square root of a negative number, and taking the logarithm of a non-positive number

The range of a function is all possible values of y (the dependent variable) There are several limitations on range

Square roots and exponential functions only give non-negative values, for example Asymptotes can graphically illustrate range limitations

In the above graph, y = –1 is a horizontal asymptote

The curves approach the line y = –1 when x stretches out to infinity and negative infinity

The function will never actually reach y = –1 Thus, -1 is not in the range of the function

In addition, the line x = 0 is a vertical asymptote The curves will approach, but never touch, the line x = 0

A composite function is the result of combining two or more functions at once f(g(x)), sometimes denoted (f g)(x), is a typical example of a composite function

The above is read as “f of g of x” The composite functions f(g(x)) and g(f(x)) are not necessarily the same


Example: Given f(x) = 3x + 2 and g(x) = x1

f(g(x)) = 3(x1 ) + 2

g(f(x)) = 2+x3

1

The only time the above two composite functions are equal is when x = –1 Inverses

An inverse is the “undo” of a function: it takes the output of a function and returns the input

The inverse of the function f(x) is denoted as f-1(x) f(f–1(x)) = (f f–1)(x) = x The domain of a function is the range of its inverse The range of a function is the domain of its inverse Not all inverses are functions

The graph of an inverse is the mirror image of the function across y = x (see graph below)

If the function f is one-to-one, then its inverse is a function

One-to-one means that no values in the function’s range appear more than once Example: the inverse of f(x) = x is a function because f only maps to each y-value

once The inverse of f(x) is f-1(x) = x

A one-to-one function (which, as a function, must by definition pass the vertical line test) passes the horizontal line test—any horizontal line placed on the graph intersects the function in at most one point


The above graph is f(x) = x2, which is a function because it passes the vertical-line test It does not, however, pass the horizontal-line test, so its inverse is not a function

You could also say that its inverse does not pass the vertical-line test To find the inverse of a function, let y = f(x), change all x’s to y’s and all y’s to x’s

Example: f(x) = x3 Let y = x3 To find the inverse, we switch the variables

x = y3 Solving for y, we get the inverse function y = 3 x Thus, f-1(x) = 3 x

Sometimes we do not need to actually find the inverse equation Example: Given that f(x) is a one-to-one function, if f(3) = 7, what is f(3) = 7, what is

f-1(7)? The answer is simply 3, since all we do in an inverse is switch the x and the y

Rational function The domain of a function includes all of its possible x-values

To determine the domain of a function, find the x-values at which the denominator equals 0 These values will be the only ones excluded from the domain Division by zero causes a mathematical error Any division by zero produces either a removable or a non-removable discontinuity

Removable discontinuities are “holes” in the graph If a factor (x – c) is in both the numerator and denominator, the two cancel

each other out, producing a “hole” at x = c Non-removable discontinuities are asymptotes

If a factor (x – c) is only in the denominator, an asymptote exists at x = c In both cases, c must be a real number


Example: What is the domain of y = 6-x+x

12-x-x2

2

?

We must factor the denominator x2 + x – 6 = (x – 2)(x + 3) The denominator has two roots, 2 and -3 Thus, the domain includes all values of x except 2 and -3

If we factor the numerator, we can find out more details about the graph

y = 3)+2)(x -x(3)+4)(x -x(

Both the numerator and the denominator have a factor of (x + 3) Thus, -3 is the location of a removable discontinuity At x = –3, a hole exists in the graph

The other root, x = 2, is the location of a vertical asymptote We cannot cancel (x – 2) out of the expression

The range of a function includes all of its possible y-values The range of rational functions can be limited by a horizontal asymptote If the exponent degree of the numerator is greater than the degree of the denominator,

there is no horizontal asymptote

Example: y = 1-x

2+x+x3

The degree of the numerator is 3 The degree of the denominator is 1 The degree of the numerator is greater than the degree of the denominator, so

there is no horizontal asymptote The range, therefore, includes all real numbers

If the degree of the numerator is the same as the degree of the denominator, then an

asymptote exists at y = bc

c is the leading coefficient of the numerator, and b is the leading coefficient of the denominator

Example: What is the horizontal asymptote of y = 43

24

x-x6x7+x3 ?

Both the numerator and the denominator have a degree of 4 c = 3 and b = –1

Remember that the leading coefficient comes before the variable with the highest power

In the denominator, the term with the highest power is –x4

The horizontal asymptote is y = 1-3 or y = –3

Note that even though y = –3 is a horizontal asymptote, it is still in the range

because f( )187- = –3

If the degree of the numerator is less than the degree of the denominator, an asymptote exists at y = 0 (the x-axis)


As x increases, the numerator will increase at a slower rate than the denominator will because the denominator has a higher exponent

Eventually, the ratio of the numerator to the denominator will approach ∞

1- ,

which effectively equals 0 The inverse of a rational function is not necessarily a function

Given rational function Q(x)

)x(P , the inverse is NOT simply P(x)

)x(Q

We must find the inverse by interchanging variables and solving for the new dependent variable (as before)

Exponential function

The independent variable in an exponential function is in an exponent

The general form of this type of function is ax The base of an exponential function, a, must be positive The domain is all real numbers The range is all positive numbers A horizontal asymptote exists at y = 0 The inverse of an exponential function is a logarithmic function Regardless of its base, an exponential function will contain the point (0, 1) if it has a

coefficient of 1 because a0 = 1


Logarithmic functions

The independent variable of a logarithmic function is in the argument of a logarithm

The general form is log(x) Log stands for a logarithm taken on base 10

Log(x) is the same as log10(x) You may also see Ln(x)

Ln stands for the natural logarithm, taken on base e e is a constant like pi e = 2.71828182846… e is important because lots of natural phenomena are based on e

Consequently, the logarithm based on e is called the natural logarithm Ln(x) is the same as loge(x) You will need to know where the Log and Ln functions are on your calculator

Logarithms are used to find the power to which a base is taken to produce the resulting argument Example: log2(8) = x

The argument is 8, and the base is 2 Solve for x if 2x = 8 23 = 8, so the power is 3 x = 3

Logarithms and exponential expressions cancel each other out when the bases are the same Example: log7(72) = x

We can rewrite this equation as 72 = 7x Thus, 2 = x

Example: ln(e 32

) = x

x=32


Special rules exist for operations on logarithms When the entire argument has an exponent, we can turn the exponent into a coefficient

of the logarithm Example 1: log A2 = 2log A Example 2: log(x3 + 7x2 – 5)2 = 2log(x3 + 7x2 – 5)

We cannot move the other exponents because they only apply to individual terms, not the entire argument

When two logarithms of the same base are added, we can combine them into one logarithm with the arguments multiplied together Example 1: log A + log B = log AB Example 2: log5(x + 2) + log5(x – 6) = log6(x+2)(x – 6) = log5(x2 – 4x – 12)

By the same token, when two logarithms of the same base are subtracted, we can combine them into one logarithm with the first argument divided by the second Example 1: log A – log B = log A/B

Example 2: log6(x + 2) – log5(x – 6) = log5(6-x2+x )

The three rules above can also be used in reverse To find a logarithm in a base other than 10 or e, use the following formula

Logbased(argument) = )baselog(

)umentlog(arg

Example: Find log6(43) Since most calculators don’t have a base-6 logarithm function, use the formula

and plug in )6log()43log(

The answer is about 2.0992 If you plug in 6 to the 2.0992 power, you get 43

The base of a logarithmic function must be positive The domain is all positive numbers The range is all real numbers A vertical asymptote exists at x = 0 The inverse of a logarithmic function is an exponential function Regardless of the base, a logarithmic function will contain the point (1, 0), provided the

argument’s coefficient is 1, because log(1) = 0

Complex Numbers Definitions

A complex number is any number in the form a + bi a and b are real numbers, and i is the imaginary unit

i = 1- or i2 = -1 All pure real numbers and all pure imaginary numbers are technically complex numbers,

with b = 0 and a = 0, respectively


Operations with complex numbers We can simplify higher powers of i

Example: Find the value of i75 We need to find the pattern to the powers of i

i1 = i

i2 = –1

i3 = –i

i4 = 1

i5 = i

We can see that the pattern repeats every 4 powers This observation gives us easy shortcut to solve i75

1. Divide 75 by 4 2. Take the remainder, ¾, and ignore the 4 in the denominator 3. Raise i to the power that you found in Step 2 4. i3 = i75= -i

Let’s try i713 1. Divide 713 by 4 2. Take the remainder, ¼, and ignore the 4 in the denominator 3. Raise i to the power that you found in Step 2 4. i1 = i713 = i

Notice that the sum of every four terms is 0 With this rule, we easily find that i34 + i35 + i36 + i37 = 0 We can also find i + i2 + i3 + … + i53 + i54

We know that i + i2 + i3 + … + i52 = 0 0 + i53 + i54 = 0 + i + (-1) = i – 1

Treat i as a variable when adding and subtracting (combine like terms) Use the distributive property when multiplying two complex numbers

In the end, simplify i2 = -1 Complex conjugates are pairs of complex numbers that come in the form a + bi and a – bi

The complex conjugate of i is -i The complex conjugate of 2 – 3i is 2 + 3i The complex conjugate of 4 is 4 because there is no imaginary part

A fraction with an imaginary expression in the denominator needs to be simplified Fix this by multiplying both the numerator and denominator by the complex conjugate

of the denominator

Example: 2i - 83i+4

We need to get rid of the i in the denominator We will multiply top and bottom by the conjugate of the denominator, 8 + 2i

2i)+2i)(8 - (82i)+3i)(8+(4 =

4+16i-16i+ 646-24i+i8+32 =

6832i+26 =

34i16+13


Complex numbers as roots of equations Any polynomial with degree n will have n (possibly nondistinct) roots among the complex

numbers All complex roots (that have nonzero imaginary parts) come in conjugate pairs

Example: If a polynomial has a root of 7 + 2i, it must have another root, 7 – 2i Since complex roots must come in pairs, then a polynomial with an odd degree must have an

odd number of real roots For a quadratic equation, the nature of the roots is determined by the discriminant of the

quadratic formula: b2 – 4ac If the discriminant is positive, both roots are real

The roots are 2a

ntdiscrimina±b-

If the discriminant is 0, the roots are real and identical

The root is 2ab-

If the discriminant is negative, the roots are complex conjugates In the quadratic equation, taking a square root of a negative discriminant creates an

imaginary unit

Reading Graphs of Functions Linear Functions

Linear functions (linear equations) are straight-line graphs These functions have x raised to the first power

By reading the graph, we can figure out the equation it represents

First, we look for the y-intercept, the value of y where the line crosses the y-axis The y-intercept in the graph above is -1

In slope-intercept form, which is y = mx + b, the y-intercept is b b = –1

We still need to find m, the slope We can read two points from the graph, (–2,0) and (0, –1)

Using the formula for slope, m = 12

12

x-xy-y , we have m =

(-2) -00--1

m = –21


Therefore, the above graph represents y = –21 x – 1

Quadratic Functions Quadratic functions (quadratic equations) are U-shaped graphs

These functions have x raised to the second power

The above graph shows a parabola that follows the standard form y = a(x – h)2 + k

Standard form is also known as vertex form because the point (h, k) is the vertex, the turning point of the parabola

To find the equation of the parabola, we must find the vertex first Since the parabola opens upwards, we look for the lowest point The lowest point is (2, -3)

Putting the vertex into the standard form equation, we have y = a(x – 2)2 – 3 To find a, we need to plug in another point

We can read from the graph the point (0, -1) –1 = a(0 – 2)2 – 3

a = 21

The graph above represents y = 21 (x – 2)2 – 3

Higher order functions Higher order functions (higher order equations) fall into two general types of graphs

If the order (degree of the highest exponent) is even, the graph will start and end on the same side of the y-axis


The above graph starts and ends on the same side, the positive side, of the y-axis,

which means the order is even

The graph shows the function y = 21 x4 + x3 – 2x2

If a test question ever asks you to find the equation from a graph like this one, eliminate the answer choices whose orders cannot possibly be correct In this case, we would eliminate all the choices with odd orders

Then, graph the remaining choices on your calculator to find the equation that matches

Alternatively, you can plug points from the graph into the remaining equations and see if they solve correctly

If the order is odd, the graph will start and end on opposite sides of the y-axis


The above graph starts on the negative side of the y-axis and ends on the positive side,

which means the order is odd The graph shows the function x5 + 2x4

Exponential functions Exponential functions create curves that have a horizontal asymptote

The above graph shows y = ex, and the asymptote is y = 0

Logarithmic functions Logarithmic functions create curves that have a vertical asymptote


The above graph shows y = ln(x), the natural logarithm of x, and the asymptote is x = 0 Notice that this graph is the inverse of the exponential graph

Flipping the exponential graph on the x = y line will yield the above logarithmic graph

Sequences, Series, and Means Arithmetic sequences

Arithmetic sequences are patterns of numbers that have a common difference d Example: 2, 5, 8, 11… Here, the difference d between consecutive terms is 3

8 – 5 = 3 11 – 8 = 3

To find the nth term of an arithmetic sequence, use the formula nth term = first term + d(n – 1) The 8th term in the example sequence above would be 2 + 3(8 – 1) = 23 It makes sense that 7 “gaps” exist between the first and the eighth terms

Arithmetic series An arithmetic series is the sum of an arithmetic sequence

The formula to find the sum of the first n terms is 2

term) last + term n(first

2

term) last + term n(first gives the average of all the terms, and multiplying the average by

n will yield the total sum Example: Find the sum of the arithmetic progression: 31, 34, 37…94, 97

First, we must find n, the number of terms in the series

The formula to find the number of terms is n = 1+d

term first - term last


If we think of the terms as fence posts separated by uniform gaps, then we would know the number of posts by adding one to the number of gaps

To find the number of gaps then, we must take the distance between the last post and the first post and divide that distance by the length of a gap

We see that the difference d = 3, and our set-up is n = 23=1+3

31 - 97

Now that we know n = 23, we can use the summation formula to find the sum of the series

Sum = 1472=2

97)+23(31

Often, summation problems will use sigma notation The Greek letter sigma is ∑

We can express the sum of the numbers 1 to 10 as ∑10

1=k

k

The index k starts at 1, the lower bound, and increases by 1 for each term until it reaches 10, the upper bound

The bounds are also called limits of summation Our expression is the same as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

Example: Find ∑12

7=n

m3

Listing the terms, we find that they are 21, 24, 27, 30, 33, 36 From m = 7 to m = 12, we have 6 terms (12 – 7 + 1 = 6)

Again, we can use the summation formula, sum = 171=2

)36+6(21

Arithmetic mean An arithmetic mean is the average of two or more numbers

Example: What is the arithmetic mean of 5, 7, 9, 11, and 13? We can solve this problem by adding up all the terms and dividing by the number of

terms

9=5

13+11+9+7+5

An alternative strategy is to recognize the terms as an arithmetic sequence The term in the middle will equal the average In this case, the middle of the five terms is 9 If the problem had an even number of terms, we would only need to average the

middle two terms to find the average of the whole sequence Given a set of unrelated numbers, of course, the arithmetic sequence approach will

not work Geometric sequence

Geometric sequences are patterns of numbers with a common ratio r Example: 1, 2, 4, 8, 16… Here, the common ratio r is 2

2=12


2=24

To find the nth term of a geometric sequence, use the formula nth term = (1st term)rn–1 Example: What is the 13th term in the sequence that begins with 1024, 512, 256, 128…?

The common ratio r is 21

1024512 =

21

512256 =

21

The 13th term is (1024)( 21 )13–1=

41

Geometric series A geometric series is the sum of a geometric sequence

The formula to find the sum of the first n terms is r-1

)r-term)(1 (first n

, where r is the

common ratio

Example: Find ∑5

1=p

p)2-(3

The first term is 3(–2)1 = –6 The second term is 3(–2)2 = 12

We can find the common ratio r by dividing: 6-

12 = –2

We know that n = 5 because the index p goes from 1 to 5

Thus, the sum is (-2)-1(-2)-6(1- 5

= 3

32)+6(1- = –2(33) = –66

Infinite series An infinite series is the sum of a pattern of numbers with an infinite number of terms

For an infinite series to be solvable, |r| < 1 If |r| ≥ 1, the series will continue to grow infinitely larger and will not approach a

sum Example: the series 2, 4, 8, 16… (r = 2) does not have a fixed sum because the

terms will simply keep getting bigger

Example: Find ∑∞

0=xx2

1

Because this series has a common ratio r = 21 , the formula we use to find the sum

will be similar to the formula for finding the sum of the first n terms of a geometric series

The formula is sum = r - 1

term first

The numerator differs from the one in the geometric series formula

Since n = ∞, (21 )∞ approaches zero, and the (1 – rn) in the geometric series

formula becomes 1


The sum is 2=2

11=

21 - 1

21

0

We were able to find a number for the sum, which means that the series converged An infinite series can also diverge

Divergent series do not add up to a nice number The harmonic series9 is a common example of a divergent series

Its progression is 11 +

21 +

31 +

41 +

51 … or ∑

∞

1=x x1

Even though the terms become smaller, they don’t scale down like the terms the convergent example, in which each term was half of the previous one

In the harmonic series, the terms keep adding up to infinity Geometric mean

A geometric mean is the product of n terms raised to n1

Example: What is the geometric mean of 3 and 27? 3 27× = 81

811/2 = 9=81 Thus, the geometric mean of 3 and 27 is 9 The answer makes sense because 3, 9, and 27 form a geometric series with a common

ratio of 3

9 The harmonic series gets its name from the way a string vibrates. The wavelengths of the harmonics (the frequencies that naturally resonate) are a half, a third, a fourth, etc. of the length of the string.


Graphing The following are graphs of various sequences and series

We can tell the above graph represents an arithmetic sequence because the terms have

equal vertical distances between each other If we connected the dots, they would form a straight line: the y-values are increasing

at a constant (linear) rate

The above graph models the series n1=n∑10

The dots no longer have equal vertical distances between each other

0

5

10

15

20

25

30

35

0 2 4 6 8 10 12

n

Arithmetic Sequence

0

10

20

30

40

50

60

0 2 4 6 8 10 12

n

Arithmetic Series


In the above geometric sequence, each term is twice as large as the previous one

This graph models the geometric series n1=n 2

4∑10

The series sums to 3.996094, and we can see that the graph approaches 4

00.5

11.5

22.5

33.5

44.5

0 2 4 6 8 10 12

n

Geometric Series

0

200

400

600

800

1000

1200

0 2 4 6 8 10 12

n

Geometric Sequence


This harmonic (infinite) series starts at 1 and keeps adding 21 +

31 +

41 +…

The harmonic series ALWAYS diverges

00.5

11.5

22.5

33.5

0 2 4 6 8 10 12

n

Harmonic Series


GEOMETRY

Right Triangles The Pythagorean theorem

The Pythagorean theorem states a special relationship that applies to all right triangles: a2 + b2 = c2 a and b are the two leg lengths of the right triangle; c

is the length of the hypotenuse We can also use the Pythagorean theorem to determine

whether a non-right triangle is acute or obtuse If a2 + b2 > c2, then the triangle is acute

c is the length of the triangle’s longest leg; a and b are the lengths of the other two lengths

If a2 + b2 < c2, then the triangle is obtuse A Pythagorean triple is a set of three integers that satisfy the Pythagorean theorem

Examples of common Pythagorean triples: 3, 4, 5 5, 12, 13 7, 24, 25 9, 40, 41 8, 15, 17

Any multiple of a Pythagorean triple also satisfies the Pythagorean theorem 6, 8, 10 is a multiple of 3, 4, 5, so it is also a Pythagorean triple

53 ,

54 , 1 is also a multiple of 3, 4, 5

Special triangles: 45-45-90 and 30-60-90 The 45-45-90 right triangle is an isosceles right triangle

Two angles are 45, and the 3rd angle is 90 The two legs are equal in length The length of the hypotenuse is always 2 times the

length of each leg Drawing a diagonal from corner to corner across a

square results in a 45-45-90 triangle


Geometry is the study of figures (both two- and three-dimensional). Of particular interest are triangles (specifically right triangles) and quadrilaterals. In this section, we will explore how to find the area and volume of such figures, in addition to several other topics.



c a

b

45°

45°

x

x

x 2


30°

60°

2x

x

x 3

The 30-60-90 right triangle is the second special right triangle The angles in this triangle measure 30, 60, and 90 The shortest side is opposite the 30 angle The length of the hypotenuse is 2 times the length of the

shortest side The length of the other leg (the leg opposite the 60° angle)

is 3 times the length of the shortest side Drawing one altitude in an equilateral triangle results in

two 30-60-90 triangles

Coordinate Geometry Lines

The midpoint of a line segment is the point equidistant from both ends Given a line segment with two end points (a, b) and (c, d), the midpoint is found by

taking the average of the two coordinates: 2

d+b,2

c+a

Slope is a line’s ratio of vertical to horizontal change

Slope can be found given any two points (a, b) and (c, d) on a line: m = xΔyΔ=

a-cb-d

If a = c, then the slope is undefined, and the two points lie on a vertical line If b = d, then the slope is 0, and the two points lie on a horizontal line

Remember this equation as “rise over run” We use the distance formula to find the distance between any two points

d = 212

212 )y-y(+)x-x( between two points (x1, y1) and (x2, y2)

d is the distance This formula is derived from the Pythagorean theorem We can use a variation of the distance formula to find the distance between two points

(x1, y1, z1) and (x2, y2, z2) in three-dimensional space

d = 212

212

212 )z-z(+)y-y(+)x-x(

Lines can be parallel or perpendicular Parallel lines are lines in the same plane that never intersect

If lines m and n are parallel, it is notated as m || n Parallel lines have the same slope

Perpendicular lines are lines that intersect to form 90 angles If lines m and n are perpendicular, it is notated as m n The slopes of perpendicular lines are negative reciprocals of each other

Example: If a line has a slope of 53 , a perpendicular line has a slope of

35-

Horizontal and vertical lines are perpendicular to each other, even though their slopes are 0 and undefined, respectively


A transversal is a line that intersects two parallel lines Vertical angles (at right, 1 and 3, 2 and 4, 5 and 7, 6

and 8) are congruent Corresponding angles (1 and 5, 2 and 6, 4 and 8, 3 and

7) are congruent Alternate interior angles (4 and 6, 3 and 5) are

congruent Alternate exterior angles (1 and 7, 2 and 8) are

congruent Consecutive angles (1 and 4, 2 and 3, 5 and 8, 6 and 7) are supplementary (add up to

180) Same-side interior angles (4 and 5, 3 and 6) are supplementary10

Properties and types of quadrilaterals A quadrilateral is a four-sided polygon

The measures of the interior angles of a quadrilateral add up to 360 The formula for the number of angles in a polygon is (n – 2)/180

A trapezoid is a quadrilateral with exactly one pair of parallel sides The parallel sides are called bases The non-parallel sides are called legs An isosceles trapezoid has congruent legs, base angles, and diagonals A right trapezoid has one right base angle

Area = )h)(b+b)(21( 21

b1 and b2 are the lengths of the bases and h is the height In a coordinate system, the two bases have the same slope (since they are parallel)

The two legs have different slopes A parallelogram is a quadrilateral with two pairs of parallel sides

Opposite angles and sides are congruent Consecutive angles are supplementary The diagonals bisect each other

To bisect means to halve an angle Area = bh

b is the length of a base and h is the perpendicular height In a coordinate system, opposite sides have the same slope and length

A rectangle is a parallelogram with four right angles All properties of parallelograms apply to rectangles Its diagonals are congruent Area = Lw, where L is the length and w is the width In a coordinate system, opposite sides have the same slope and length, and the slopes of

adjacent sides must be perpendicular A rhombus is a parallelogram with four congruent sides

All properties of a parallelogram apply to a rhombus Its diagonals are perpendicular to each other

10 The Princeton Review sums all of the above stuff really nicely in “Fred’s Theorem”: all the small angles are congruent. All the big angles are congruent. A small angle and a big angle are supplementary. – Dean

b2

h

1 2

3 4

5 6

7 8

b1


The diagonals bisect each other The diagonals bisect the corner angles to form 4 congruent right triangles

Area = 21dd21

d1 and d2 are the lengths of its two diagonals In a coordinate system, the slopes of the diagonals are negative reciprocals

of each other (because they are perpendicular) A square is a parallelogram that is both a rectangle and a rhombus

All properties of rectangles and rhombuses apply to squares The diagonals form 4 congruent isosceles right triangles Area = s2

s is the length of one side The diagonals are perpendicular, bisect each other, and have the same length

Congruency and Similarity Congruence

Two figures are congruent if they have the same shape and area In other words, congruent figures have sides and angles of the same measures The following figures are all congruent

The triangle is rotated and flipped several different ways, but the figure’s shape and

area remain the same Similarity

Two figures are similar if they have the same shape The following figures are all similar


The ellipses are different sizes, but they all have the same shape

Plane and Solid Figures Area of triangles, quadrilaterals, and circles

There are several formulas that allow us to find the area of a triangle

A = bh21

b is the length of the base and h is the height; A is area Heron’s formula11: A = c)-b)(s-a)(s-s(s

a, b, and c are the lengths of the three sides of the triangle; s = 2

c+b+a

A = Csinab21

a and b are two sides and C is the measure of the angle between these two sides Formulas for finding the area of quadrilaterals vary depending on the type of quadrilateral in

question These formulas can all be found in the previous section (“Coordinate Geometry:

Properties and types of quadrilaterals”) A circle is the two-dimensional set of all points equidistant from one center point

A = πr2 r is the radius of the circle

A “sector” of a circle is visually analogous to a slice of pie

If you have the arc measurement in degrees, A = πr2 360

measurearc

“Arc measure” is the degree measure of the “crust” of the sector slice

If you have the arc measurement in radians, A = πr2 radians

measurearc

Area of regular polygons Regular polygons have sides of equal length and angles of equal size

We can divide these polygons into isosceles triangles, with each side of the polygon as a base

11 This formula is notoriously difficult to type into calculators. Be careful. – Steven


We can then find the area of each isosceles triangle and multiply it by the number of triangles to find the area of the whole polygon

Example: Find the area of a regular heptagon with a side length of 4 and an apothem length of 4.153 A heptagon (sometimes called a septagon) has 7 sides

The side length is 4, so the base of the isosceles triangle is 4 An apothem is the distance from the center of a regular polygon to the middle of a

side Apothems are always perpendicular to the sides In the drawing above, the apothem is the height of the triangle Later, you will be able to use trigonometry to find the height of the triangle For now, the height (apothem) is given as 4.153

Using the formula for the area of a triangle, A = bh21

, we have

A = 306.8=)153.4)(4(21

The heptagon has 7 sides and, therefore, 7 isosceles triangles, so we need to multiply the area of the triangle by 7 The area of the heptagon is A = 8.306 × 7 = 58.142

Area and volume of prisms, pyramids, cylinders, spheres, and cones A prism consists of two parallel and congruent bases and the space between the two bases12

Surface area = area of the 2 bases + area of lateral faces (for our purposes, the lateral faces are rectangles)

Volume = (area of a base)(height) A pyramid is akin to a prism, but it has one base instead of two

This base rises up to a vertex (point of intersection of the sides) SA = area of the base + area of the lateral faces (for our purposes, triangles)

12 This is actually the definition of a regular prism. Most basic math (including Decathlon math), however, focuses almost exclusively on regular prisms, so we will, too.


Volume = ( 31

)(area of the base)(height)

A cylinder is essentially a circular prism SA = 2πr2 + 2πrh

r is the radius of a base, and h is the height of the cylinder Volume = πr2h

A sphere is the three-dimensional set of all points equidistant from one center point SA = 4πr2

r is the radius of the sphere

Volume = 34

πr3

A cone is a pyramid with a circular base

SA = πr2 + πr 22 h+r r is the radius of the base, and h is the height

22 h+r is the lateral height, the distance from the edge of the base to the top point

Volume = 31

πr2h

Properties of similar figures Corresponding parts of similar figures are proportional There are a few ways to test triangles for similarity

SSS similarity theorem: if two triangles exist such that all three pairs of corresponding side lengths form a constant ratio, then the two triangles are similar

SAS similarity theorem: if two triangles exist such that two pairs of corresponding side lengths form a constant ratio and the angles included between those sides are congruent, then the two triangles are similar

AA similarity theorem: if two triangles exist such that two pairs of corresponding angles are congruent, then the triangles are similar

These theorems can be extended to other geometric figures, too If all the corresponding angles in two figures are congruent, then the two figures are

similar Properties of circles

Angle measures are an important part of circle geometry A circle has 360 or 2 radians

radians = 180 Example: How many degrees is 1 radian?

(1)( π

180 ) = 3.57

The measure of a central angle is equal to the measure of the intercepted arc The measure of an inscribed angle is equal to the half the measure of the intercepted arc The measure of an angle in the interior of the circle is half the sum of the two

intercepted arcs (see circle diagrams on the last page of this section) The measure of an angle in the exterior of the circle is half the difference of the two

intercepted arcs Tangents, secants, and chords are the main three types of lines associated with circles


A tangent is a line that intersects the circle at only one point Tangent lines “touch” circles Tangents are perpendicular to the radius drawn to the point of tangency

A secant is a line that intersects the circle at two points Secant lines go through circles

A chord is a line segment whose two endpoints lie on the rim of the circle The longest chord in a circle is the diameter If two chords are the same distance from the center of a circle, they are congruent

Their intersected arcs are also congruent If two chords are congruent or if their intersected arcs are congruent, the two chords

are the same distance from the center of the same circle Chord-Chord Power Theorem: two intersecting chords form four line segments such

that the product of one chord’s line segment lengths equals the product of the other chord’s line segment lengths (see circle diagrams below)

Secant-Tangent Power Theorem: the product of the lengths of the secant and its external part is equal to the square of the length of the tangent

Secant-Secant Power Theorem: the product of the lengths of one secant and its external part is equal to the product of the lengths of the other secant and its external part


O

NH

C

U

S

O

L

Y

D

O

H

U

ES

Two tangents from a common exterior point are congruent; in this case, DL DY

Any radius drawn to a point of tangency is perpendicular to the

tangent; in this case, DLOL⊥

and DYOY⊥

If a tangent and a secant are drawn from a common exterior point, the product of the secant’s length and the length of its external part equals the square of the length of the tangent; in this case,

2(HE)=SE×UE [Secant-Tangent Power Theorem]

If two secants are drawn from a common point, the product of the first secant’s length and the length of its external part equals the product of the second secant’s length and the length of its external part:in this case, SH×SN=SU×SC[Secant-Secant Power Theorem]

m S = ( )UHm-CN∠m)(21

∠

Two intersecting chords form four line segments such that the product of one chord’s line segments equals the product of the other chord’s line segments; here, BP x PI = DP x PE.

[Chord-Chord Power Theorem]

)BEm=DIm)(21(=BPE∠m ∠∠

PD

B

I

E

O


TRIGONOMETRY

Right Triangle Relationships In a right triangle ABC where C is the right angle

sine of an angle = hypotenuse

opposite

cosine of an angle = hypotenuse

adjacent

tangent of an angle = adjacentopposite 13

Examples

sinA = cosB = ca

sinB = cosA = cb

tanA = cotB = ba

tanB = cotA = ab

secA = cscB = bc

secB = cscA = ac

csc is the reciprocal of sin sinC = 1 cscC = 1

sec is the reciprocal of cos cosC = 0 secC is undefined

cot is the reciprocal of tan cotC = 0 tanC is undefined

13 An easy way to remember these three properties is with the mnemonic “SOH-CAH-TOA.”


Trigonometry is the study of angles and the angular relationships of planar figures. The trigonometric functions are also called the circular functions because they can all be derived from the unit circle.

According to the USAD outline, 7 questions (20% of the test) will come from this section



Trigonometric Functions Trig functions and quadrants

The sign of the value of a function depends on the quadrant of the angle All three main functions (sine, cosine, tangent) are

positive in Quadrant I Sine is positive in Quadrant II Tangent is positive in Quadrant III Cosine is positive in Quadrant IV14

Each of the three reciprocal functions (cosecant, secant, and cotangent) is positive in the same quadrants as its corresponding “main” function

Trig functions and reference angles We can use the reference angle to determine the value

of a trigonometric function If the angle is in Quadrant I, is the reference angle

Example: 60 is in Quadrant I, so its reference angle is 60 If the angle is in Quadrant II, 180 – θ (or π ) is the reference angle

Example: 4π3 is in Quadrant II, so its reference angle is

4π=

4π3-π

If the angle is in Quadrant III, θ – 180 or π is the reference angle Example: 200 is in Quadrant III, so its reference angle is 200 – 180 = 20

If the angle is in Quadrant IV, 360 – θ or 2π – θ is the reference angle

Example: 3π5 is in Quadrant IV, so its reference angle is

3π=

3π5-π2

When using reference angles, follow the ASTC rule mentioned above to put the correct sign on the result

Example: Find cos( )3π4

3π4 is in Quadrant III, so its reference angle is

3π=π-

3π4

cos(21=)

3π

In Quadrant III, tangent is positive, and sine and cosine are negative

Thus, cos(21-=)

3π4

Inverse Trigonometric Functions Basic information

The inverse trig functions include arcsin, arccos, arctan, arccsc, arcsec, and arccot Basically, if sinA = B, then arcsinB = A

Similar relationships apply for the other inverse functions as well sin-1A is the same as arcsinA

14 My Algebra II teacher taught me a trick to remember this. If you go in order from quadrants I to IV, the order of positive functions is all functions, sine, tangent, and cosine. All students take classes. – Dean

A S

T C


All inverse trig functions can be notated either way Evaluating inverse trig expressions

Substitution can be a powerful tool in evaluating inverse trig functions

Example: to evaluate cos(arcsin( ))21 , let θ =arcsin

21

Now, we’re just trying to solve cosθ

arcsin2π=

21

and cos

23=

6π

Notice that we use just the principal value of arcsin

Otherwise, cos23-=

6π5 would also be an answer

The domains and ranges of inverse trig functions Trig functions don’t pass the horizontal line test, so their inverses are not functions To be able to work with the inverse functions as functions, we must limit their domains and

ranges (see table below) These limitations ensure that the inverse functions pass the vertical line test

Inverse Trig Functions

Function Domain Range

Arcsin [– 1, 1] [2

,2

-ππ

]

Arccos [– 1, 1] [0, ]

Arctan ( , ) (2π,

2π- )

Arccsc (– ,–1] 1, ) π π

[ ,0) (0, ]2 2

Arcsec (– ,–1] 1, ) π π

π[0, ) ( , ]2 2

Arccot ( , ) (0, )

Graphs Period

The period of a function is the interval over which it repeats All trigonometric functions are periodic

Sine and cosine (and their reciprocal functions) have periods of 2 Tangent and cotangent have periods of The periods of sine and cosine (and their reciprocal functions) can be determined by the

coefficient of the angle (here, x)

Example: the period of sin(kx) is kπ2


Example: the period of tan(kx) is kπ

Amplitude The amplitude of a cyclical function is half the distance between the maximum and

minimum height of a wave Since amplitude measures distance, it is always positive

Sin and cos have amplitudes that can be determined by the coefficient of the function Example: the amplitude of kcosx is |k|

The other functions don’t really have an “amplitude” because their range is unbounded, but the coefficient can stretch the graph vertically

Horizontal shifts A constant term inside the function can horizontally shift a function’s graph

Example: the horizontal shift of sec(x – k) is k to the right Note that the shift is positive (to the right) even though the coefficient ( – k) is

negative If the function were sec(x + k), the shift would be negative (to the left)

Vertical shifts A constant term outside the function can vertically shift the function’s graph

Example: the vertical shift of sin(x) + k is k upward Note that this shift is positive shift If the function were sin(x) – k, the shift would be negative (down)

Combining all these properties

Example: f(x) = 3cos(7x + 1-)2π7

The first thing we need to do is factor out the coefficient attached to the x

f(x) = 3cos[7(x + 1-)]2π

Only when x is by itself can we find the period and horizontal shift

This function has a period of 7π2 , an amplitude of 3, a shift of

2π to the left, and a shift

of 1 down

period = 2C

amplitude = B

amplitude = B

*

*phase displacement = DC

vertical shift = A

Graph of Bsin(Cx + D) + A


Identities Purpose

Oftentimes, problems with trig functions in them will not be solvable as presented You’ll have to convert functions using the identities below to solve the problem

Reciprocal identities

sin x = xcsc

1 xcsc = xsin

1

cos x = xsec

1 xsec

=

xcos1

tan x = xcot

1 xcot

=

xtan1

Quotient identities

tan x = xcosxsin =

xcscxsec

cot x = xsinxcos =

xsecxcsc

Pythagorean identities sin2x + cos2x = 1 tan2x + 1 = sec2x 1 + cot2x = csc2x

Sum identities sin(x + y) = (sinx)(cosy) + (cosx)(siny) cos(x + y) = (cosx)(cosy) + (sinx)(siny)

tan(x + y) = )y)(tanx(tan-1

ytan+xtan

Difference identities sin(x – y) = (sinx)(cosy) – (cosx)(siny) cos(x – y) = (cosx)(cosy) + (sinx)(siny)

tan (x – y) = )y)(tanx(tan+1

ytan-xtan

Double angle identities sin(2x) = 2(sinx)(cosx) cos(2x) = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1

tan(2x) = xtan-1

xtan22

Half angle identities

2

xcos-1±=)2xsin(

2

xcos+1±=)2xcos(

xcos+1xcos-1±=)

2xtan(

Phase identities

sinx = )x-2πcos(


cosx = )x-2πsin(

Odd/even properties sin(–x) = –sinx csc(–x) = –cscx tan(–x) = –tanx cot(–x) = –cotx cos(–x) = cosx sec(–x) = secx

Sum-to-product identities

)2

y-xcos()2

y+xsin(2=ysin+xsin

)2

y-xsin()2

y+xcos(2=ysin-xsin

)2

y-xcos()2

y+xcos(2=ysin-xsin

)2

y-xsin()2

y+xsin(2-=ycos-xcos

Product-to-sum identities

2

)y-xsin(+)y+xsin(=ycosxsin

2

)y-xsin(+)y+xsin(=ycosxsin

2

)y+x(osc-)y-xcos(=ysinxsin

Trigonometric Equations Law of Sines

The law of sines states that in a triangle, the ratio of the sine of an angle to the length of the opposite side is the same for all three angles

aAsin =

bBsin =

cCsin

As long as we have one angle-side pair (A and a, B and b, or C and c) and another side or angle, we can find the rest of the variables

A

BC

c

a

b


Law of Cosines The law of cosines is a general form of the Pythagorean theorem

Whereas the Pythagorean theorem only works for right triangles, the law of cosines works for any triangle

Given two sides and the angle between them, we can find the length of the third side (refer to the above triangle for the following formula)

c2 = a2 + b2 – 2ab(cosC) In a right triangle, c is the hypotenuse, which means C is the right angle

The cosine of ninety degrees is 0, which is why the last term in the formula disappears in the Pythagorean theorem

Algebraic equations involving trig functions Unless there are restrictions on domain and range, an infinite number of possible solutions

exist to a trigonometric equation To solve for all solutions, remember that the functions are periodic

If x is a solution, then 360 + nx, where n is an integer, is also a solution For tangent and cotangent, 180 + nx is also a solution

The period of these functions is only 180 Check for other solutions

Example: if a solution to a sine equation is found in Quadrant I, then there should also be a solution in Quadrant II, since sine is positive in Quadrants I and II

To solve trig equations, isolate the trigonometric expression Change all trigonometric expressions to the same function

Example: cos2x + sinx + 1 = 0 First use a Pythagorean identity to convert all the expressions to sine Thus, (1 – sin2x) + sinx + 1 = 0

Use substitution if necessary Example: 2sin2x + sinx – 1 = 0

Let u = sinx Substitution and factoring give (2u – 1)(u + 1) = 0

The solutions can be found by solving u = sinx = 21 and u = sinx = –1

Then solve for x x = 30° and x = 270°


CALCULUS

Basic Limits and Continuity Limits

A limit is the y-value that a function approaches when getting infinitely close to a given x-value

Notation: )x(flimcx →

means taking the limit of f as x approaches c

c is the value that x is approaching The function does not necessarily have to be defined at c

The function can approach or at c If the function is undefined at c, check for removable discontinuities

Factor the numerator and denominator and cancel factors if possible, then try substituting c again

If the left-hand and right-hand limit of a function as x approaches c are not the same, the limit does not exist

The limit can be evaluated by substituting c as x in the function if the function exists at c Continuity

A function is continuous at c if the limit at c equals the function’s value at c In other words, it’s continuous if )x(flim

cx→= f(c)

All polynomials are continuous everywhere Rational functions may have discontinuities at vertical asymptotes or removable

discontinuities Boundaries and endpoints of piecewise functions are other possible points of discontinuity

L’Hopital’s rule15 After plugging in c, if a limit is indeterminate, we can use L’Hopital’s rule to convert the

limit into a determinate one

Indeterminate limits come in the form of 00 and

This rule will be covered after the next section on first and second derivatives.

15 Also spelled “L’Hospital,” this rule is “derived” from its common usage in infirmaries and clinics across France. Just kidding. – Sophy


Ideas related to calculus have been around since Archimedes, but it was through the independent work of Newton and Leibniz that modern calculus was developed.




First Derivatives, Second Derivatives, Antiderivatives, and Their Graphical Interpretations Derivatives

Finding a derivative is known as differentiation

The derivative of f(x) at c is defined as h

)c(f-)h+c(flim=c-)h+c(

)c(f-)h+c(flimcxcx →→

A derivative is a rate of change The formula above attempts to find how fast f(x) changes between x = c and x = c + h,

where h is so small a change that it is almost 0 Imagine hitting the accelerator in a car and trying to figure out how much the car’s

speed changed over the first millisecond The derivative describes the rate of change of the dependent variable, f(x), with respect to the

independent variable, x The derivative at a point is like the “slope” of the function at that specific point

Graphically, it is the slope of straight line that is tangent to the graph at that point

Notation: if y = f(x), then )x('f=dxdy

Formula to find derivatives of x raised to a power: 1-pp px=)x(dxd

Formula for the derivative of a constant (c): 0=)c(dxd

Constants are really cx0 Thus, the derivative of any constant is 0

Prior to 2005, the only derivatives tested at competition were polynomial derivatives In 2005, the Product Rule and Chain Rule were tested, as well as the derivatives of

trigonometric functions See the tables at the end of this section for formulas for these derivatives

The derivative of a sum is the sum of the derivatives To find the derivative of a polynomial, just take the derivative of each term Example: f(x) = 3x2 + 5x + 12

f'(x) = 6x + 5 First derivative

The first derivative is the slope of a function at a specific point The first derivative taken with respect to time is the velocity of the function at that point

Displacement is the distance of a point from its starting point The first derivative can reveal details about the graph of the function, such as maxima and

minima (see below) The first derivative can also give the slope of any point on the function, thus providing a

method of finding the tangent line at that point (see below) If the first derivative is positive in a region, then the function is increasing If the first derivative is negative in a region, then the function is decreasing If the first derivative is zero, the function is not changing at that point

The point could be a maximum or a minimum (more details later) Second derivative

The second derivative is the slope of a function’s first derivative at a specific point


The second derivative of displacement taken with respect to time is the acceleration of the function at that point

The second derivative can reveal details about the graph of the function, such as concavity and points of inflection If the second derivative is positive in a region, then the function is concave up If the second derivative is negative in a region, then the function is concave down

A point of inflection is a point in the graph of a function where the function’s concavity

changes (from up to down or down to up) The zeros of the second derivative are possible points of inflection

Antiderivative An antiderivative is a possible function that has a known derivative

Antiderivatives are also called integrals Finding an antiderivative is called integration To integrate, we reverse the steps of differentiation The integration symbol is Example: If f’(x) = 3x2 + 5, what is a possible function for f(x)?

Our integration problem is f(x) =∫ dx)5+x3( 2 The dx at the end of the expression simply shows that the argument inside the

integration is a derivative Because the two terms are added, we can split them into two integrations

f(x) = ∫∫ dx5+dx)x3( 2

We can split integrations when the terms are added or subtracted but not when the terms are multiplied or divided

The first term in the derivative is 3x2 Step 1. Add one to the power Step 2. Next, divide the coefficient by the answer you found in Step 1

The answer that you find in Step 2 is the coefficient of the antiderivative Step 3. Finally, add C

The C at the end is a constant Remember, constants differentiate to 0 We have to put the C at the end of the integral because the antiderivative

could have a constant term So, the antiderivative of 3x2 is x3 + C

Concave up

Concave down


Let’s check: if we differentiate x3 + C, we do indeed get 3x2 Then we integrate the second term, 5

The term can also be written 5x0 Its integral must have a power of 0 + 1 = 1

The coefficient of its integral is 5=15

Thus, the antiderivative of 5 is 5x + C Again, we have to put the C on the integral to account for a possible constant

Putting the terms back together, we have f(x) = x3 + C + 5x + C We can combine the unknown constants together CAUTION: the sum of the C’s is NOT 2C The C’s are constants, not variables They will combine into one unknown constant16 The final form of the antiderivative is f(x) = x3 + 5x + C This type of integral is called an indefinite integral

The answer includes a set of possible functions because the value of C is indefinite

Definite integrals produce a value because they have boundaries

Definite integrals come in the form dx)x('f∫b

a

The limits of integration are a and b After finding f(x), we find the difference between the boundaries

dx)x('f∫b

a

= f(x)| ba = f(b) – f(a)

Example: dx5+x3 25

1∫

The integral is the same as the example above except for the limits of integration Originally, we found that the antiderivative is f(x) = x3 + 5x + C Now, we have to plug in the limits and find the difference

f(5) = (5)3 + 5(5) + C = 150 + C f(1) = (1)3 + 5(1) + C = 6 + C f(5) – f(1) = 150 + C – (6 + C) = 144

Definite integrals always cancel out the C Regardless of what value C may be, we know that it is the same in both f(b) and

f(a) For this reason, the subtraction always gets rid of C

L’Hopital’s rule17 As mentioned earlier, if a limit is indeterminate, we can use L’Hopital’s rule to convert the

limit into a determinate one L’Hopital’s rule takes the derivative of the numerator and denominator (separately)

After the derivatives, we plug in c again to see if the limit has become determinate

16 In other words, one unknown number plus another unknown number equals a third unknown number. 17 Also spelled “L’Hospital,” this rule is “derived” from its common usage in infirmaries and clinics across France. Just kidding. – Sophy


The next topic will give a more detailed explanation of derivatives Here is a basic description of how to take a derivative

Suppose you have a term 4x3 Take the exponent, 3, and multiply it by the coefficient, 4

The product is 12, which becomes the coefficient of the derivative Then subtract one from the exponent

The exponent of the derivative is 3 – 1 = 2 The derivative of 4x3 is 12x2

Example: 3-x9-xlim

2

x→3

If we plug in 3, we get 0 on the top and the bottom, which is an indeterminate form We will take the derivative of the numerator and the denominator

Our limit becomes 1x2lim

x→3

Plugging in 3 again, we find that the limit is 6 If the limit still yields an indeterminate answer after the application of L’Hopital’s rule, use

the rule again (and so on until you reach a definite answer)

Graphing By looking at graphs of derivatives, we can gain information about their original function Below are three graphs: a function, its derivative, and its second derivative

The graph on the left, the parabola, is the original function, 2x23

The graph in the middle, the sloped line, is the first derivative, 3x This graph starts negative and ends positive, with a critical point at 0

The change in sign lets us know that the original function decreases until x = 0 and then increases

x = 0 is the location of a minimum At this point, the original function’s slope changes from negative to positive

The graph on the right, the horizontal line, is the second derivative, 3 This graph stays positive over its entire domain

We can tell that the original function is concave up over its entire domain


Equation of a Tangent Line Definition

The tangent line at a point has the same slope as the function at that point Finding the tangent line

First, evaluate the function at the given point to find (x1, y1) Then, evaluate the function’s first derivative at x to get the function’s slope

Now we know m at that point Next, use the point-slope formula of a line to find the equation of the tangent line

Its equation will be in the form of y – y1 = m(x – x1) Example: Find the line tangent to y = 2x3 + 4x at x = 2

The y-value at x = 2 is 2(2)3 + 4(2) = 24 The derivative is y’ = m = 6x2 + 4

m = 6(2)2 + 4 = 28 y – 24 = 28(x – 2)

y = 28x – 56 + 24 The tangent line is y = 28x – 32

Rates of Change Definitions

Rate of change is the rate at which one variable changes with respect to another variable under certain conditions

Problems involving rates of change with two variables are often called related rate problems Solving single variable problems

Single variable rate of change problems usually involve displacement, velocity, and acceleration

Example: If the velocity of a rocket is defined as v(t) = 300t2 + 20t + 100, where t is time, what is the acceleration of the rocket when t = 4? Notice that the function only has one variable, t Acceleration is the derivative, or the rate of change, of velocity

Thus, we need to take the derivative of the velocity function to arrive at the acceleration function

a(t) = v’(t) = 600t + 20 a(4) = 600(4) + 20 = 2420

Solving related rate problems A related rate word problem usually sets up a situation

A typical problem might concern an inflating balloon or a plane in flight It will give at least one rate

This rate might be the rate at which the radius of the balloon is increasing as the balloon inflates or the speed of the plane

Initial parameters will set up the problem In our examples, this parameter could be the radius of the balloon at a certain time or the

distance the plane is from an observer at a certain time The problem will ask the rate something else is changing given the above parameters

For example, it might ask the rate at which the volume of the balloon is changing at t = 5 seconds


In another example, the problem might ask the rate at which the plane’s distance from the observer is changing at v = 50 meters per second

Drawing a picture will help us set the problem up Establish relationships between the given information and what we are supposed to find to

solve the problem

For our balloon, 3rπ34=V relates the volume of the balloon to its radius

For our plane, (distance from observer)2 = (horizontal distance to plane)2 + (vertical distance to plane)2

Implicitly differentiate the equation with respect to time to get a relationship between the rate we want to find and the rate we are given

dtdV =

dtdrrπ4 2 gives the relationship between the rate of volume change and the rate of

radius change of the balloon

(2)(dist.) 0+)dt

.)dist.horiz(d)(planeto.dist.horiz)(2(=

dt)observerfrom.dist(d

Note that the derivative of the vertical distance component is 0 because it does not change

Substitute the known values and solve for the unknown rate An example

A 13 ft ladder is leaning against a wall. The ladder is sliding down the wall at a rate of 2 ft per second. How fast is the bottom of the ladder moving along the ground when the bottom of the ladder is 5 ft from the wall?18

Physical relationship: set x as the distance from the bottom of the ladder to the wall, y as the distance from the top of the ladder to the ground, and l3 as the length of the ladder Using the Pythagorean theorem, we establish the

equation x2 + y2 = 132 = 169 ft

Implicit differentiation: 2xdtdx + 2y

dtdy = 0

Given rate: dtdy = –2 ft/second

Initial conditions: x = 5 ft, and (from the Pythagorean theorem) y = 12 ft

Solve: (2)(5ft)( dtdx ) + (2)(12 ft)( –2 ft/second) = 0

dtdx = 4.8 ft/second

18 The trick answer choice would be 2 ft/second. Don’t fall for it.

13 y

x


Maxima and Minima Absolute extrema

The absolute maximum of a function is the maximum y-value it reaches It is also known as the global maximum

The absolute minimum of a function is the minimum y-value it reaches It is also known as the global minimum

Relative extrema The relative maximum of a function is a point whose y-value is greater than those of the

surrounding points Relative maxima are located at the very peak of “hills” in a graph They are also known as local maxima

The relative maximum of a function is a point whose y-value is less than those of the surrounding points Relative minima are located at the very bottom of “valleys” in a graph They are also known as local minima

Relative extrema do not necessarily have to exist Linear functions, for example, have no relative extrema

Finding absolute extrema If the function is defined over a closed interval19 and is continuous over that interval, first

plug in the one or two given endpoints Whether or not the function has a specific domain, proceed to take the first derivative of the

function Find all points where the derivative is 0 or undefined

To do so, set the derivative equal to 0 and, if the derivative includes variables in denominators, set the denominators equal to 0

These points are known as critical points Plug all critical points into the original function Compare all of the y-values generated by plugging in the endpoints and critical points

The highest y-value is the function’s absolute (or global) maximum The lowest y-value is the function’s absolute minimum

Finding relative extrema First, take the first derivative of the given function

Find all critical points There are two ways to proceed from here

The first option is to use the first derivative test The second is to use the second derivative test

The first derivative test involves examining changes in the sign of the first derivative If the first derivative changes from negative to positive around a critical point, that point

is a minimum If the first derivative changes from positive to negative around a critical point, that point

is a maximum If the sign of the first derivative does not change around a critical point, then that point

is not an extreme

19 Meaning the domain has to have two inclusive endpoints.


The second derivative test involves examining the sign of the second derivative at a critical point First, take the second derivative of the function Then, plug the critical point(s) from the first derivative into the second derivative

equation If f''(x) at a critical point is positive, that point is a relative minimum If f''(x) at a critical point is negative, that point is a relative maximum If f''(x) at a critical point is zero, then there is no extrema at that critical point

Max/min word problems Word problems often involve some sort of optimization (making some quantity the

biggest/smallest) under some kind of constraint To solve: write down an equation for the quantity you’re maximizing/minimizing, take the

derivative, find the critical points, and then test those points out Example: A farmer has 20 ft of fence and wants to have a rectangular fence that encloses the

largest possible area. What should the dimensions of his fence be? The constraint is given by the perimeter: for length L and width w, 20 = 2L + 2w

The quantity we’re trying to optimize is area: A = Lw Through substitution, we can rewrite the equation for area in terms of one variable: A =

(L)(10 – L) Now we can take the derivative: A’ = 10L – 2L Setting the derivative equal to 0 yields 10 – 2L = 0, or L = 5 ft L = 5 ft (and w = 5 ft) will give us the largest fence in terms of area enclosed

That the length and width are equal is no surprise: when the 4 sides of a rectangle are limited to a specific perimeter, squares maximize area

When only 3 sides of a rectangle are limited to a set perimeter, however, squares will not maximize area

Example: A farmer has 20 ft of fence and wants to build a rectangular pigpen that encloses the largest possible area. He will build the pen next to his 20 ft-long barn, which will provide one side of the pen. What should the dimensions of his pen be? The area formula remains the same: A = Lw The perimeter formula changes, since only one length and two widths are limited

20 = L + 2s L = 20 – 2w Only one length is limited because the barn is 20 ft long The farmer has only 20 ft of fence, so he would not be able to build past the

length of the barn We substitute L in the area formula to get A = (20 – 2w)(w)

A = 20w – 2w2 Now we take the derivative to find the maximum

A’ = 20 – 4w 0 = 20 – 4w w = 5

Plugging w = 5 into the perimeter equation, we find that L = 10 Notice that the maximum area was not achieved by creating a square


Enrichment: Other Derivatives

THE CURRICULUM OUTLINE IS VAGUE ABOUT “DIFFERENTIAL CALCULUS,” BUT THESE HAVE BEEN TESTED IN COMPETITION

Derivative of sine functions: )'u)(ucos(=))u(sin(dxd

x

Derivative of cosine functions: )'u)(usin(-=))u(cos(dxd

Derivatives of exponential functions: dxd

(eu) = (eu)(u’)

Derivatives of logarithmic functions: dxd

(ln(u)) = )'u)(u1(

Chain Rule for differentiation: dxd

[f(g(u))] = f’(g(u))g’(u)

Product Rule for u and v as functions of x: (uv)’ = (u’)(v) + (u)(v’)

Quotient Rule for u and v as functions of x: )'vu

( = 2v

'uv='vu

Enrichment: Still More Derivatives

THESE DERIVATIVES MAY OR MAY NOT BE TESTED AT COMPETITION

dxd

(tan(u)) = (sec2(u))(u’) dxd

(sin-1(u)) = 2u-1

'u

dxd

(cot(u)) = -(csc2(u))(u’) dxd

(cos-1(u)) = –2u-1

'u

dxd

(sec(u)) = (sec(u))(tan(u))(u’) dxd

(tan-1(u)) = 2u+1

'u

dxd

(csc(u)) = -(cot(u))(csc(u))(u’) dxd

(cot-1(u)) = – 2u+1

'u

dxd

(au) = ln(a)(au)(u’) dxd

(sec-1(u)) = 1-u|u|

'u2

dxd

loga(u) = ()aln(

1)(

u1

)(u’) dxd

(csc-1(u)) = –1-u|u|

'u2


POWER LISTS

TERMS – GENERAL MATH

Arrangement principle To find the total number of arrangements of n objects where r objects are indistinguishable, divide the total number of arrangements by r!:

!r!n

Combination An arrangement of a collection of objects in which order does not

matter; )!r-n)(!r(

!n=Crn

Factorial The product of a non-negative integer n with all of the positive integers less than n; this is expressed as n!

Multiplication principle To find the total number of possibilities when picking one each of several different objects (each with several choices), multiply the total number of choices for each object

Percentage Represents100

n of the whole

Permutation An arrangement of a collection of objects in which order matters;

r)!-n(!n=Prn

Probability The chance that a given event will happen; equal to the number of outcomes in which the event occurs divided by the total number of possible outcomes

TERMS – ALGEBRA

Absolute value The non-negative value of a number; in other words, how far a number is from 0 on the number line

Arithmetic sequence A pattern of numbers that have a common difference

Arithmetic series The sum of an arithmetic sequence

Arithmetic mean The average of two or more numbers

Asymptote A line that a function approaches but never reaches

Complex conjugate A pair of complex numbers in the form a + bi and a – bi

Complex number Any number in the form a ± bi where a and b are real numbers and i is the imaginary unit

Composite function A function resulting from using one function as the input of another

Convergent Applies to an infinite series which approaches a fixed sum (|r| < 1)

Degree The highest exponent power of a polynomial; also known as order

Discriminant In the quadratic formula, the part under the square root; b2 = 4ac


Divergent Applies to an infinite series which does not approach a fixed sum (|r| ≥ 1)

Domain All possible values for the independent variable (often x) in a function

Equation A mathematical statement that two expressions are equal

Exponential function A function in which the independent variable is an exponent

Function An equation in which each possible value of the independent variable corresponds to one and only one value of the dependent variable

Geometric sequence A pattern of numbers that have a common ratio

Geometric series The sum of a geometric sequence

Geometric mean The product of n numbers to the power of (1/n)

Higher order equation An equation in which the highest power of the variables is greater than 2

Horizontal line test An equation passes this test if a horizontal line intersects its graph at no more than one point; if a function passes this test, its inverse is also a function

Inequality A mathematical statement that two expressions are unequal

Inverse The “undo” of a function; takes the output of a function and returns the input

Infinite series The sum of a pattern of numbers with an infinite number of terms

Linear equation An equation in which the highest power of the variables is 1

Logarithmic function A function in which the independent variable is in the argument of a logarithm

Parabola The U-shaped graph of a quadratic equation

Perpendicular line A line with a slope that is the opposite reciprocal of the slope of another line

One-to-one function A function in which none of the values of its range repeats more than once

Order See degree

Quadratic equation An equation in which the highest power of the variables is 2

Range All possible values for the dependent variable (often y) in a function

Root A number that yields zero when plugged into an expression; also known as an x-intercept and as a zero of an equation

Vertex The turning point of a parabola

Vertical line test An equation passes this test if a vertical line intersects its graph at no more than one point; if an equation passes this test, it is a function

X-intercept See root

Y-intercept The point where an equation intercepts the y-axis; equal to b in the slope-intercept form of a line (y = mx + b)

Zeros (of an equation) See root


FORMULAS AND THEOREMS – ALBEGRA

Difference of cubes formula (x3 – y3) = (x – y)(x2 + xy + y2)

Factor theorem If f(x) is a polynomial and f(c) = 0, then (x – c) is a factor of f(x); is the special case of the remainder theorem

FOIL Stands for “first, outer, inner, last”; a quick way to convert the factored out form of a quadratic back to ax2 + bx + c form

Point-slope formula y – y1 = m(x – x1)

Quadratic formula x =

2a(4ac)-b±b- 2

Rational roots theorem Given a polynomial of the form axn + … + c, all of the rational real

roots will come in forms like pq± , where p represents all the factors of

a and q represents all the factors of c

Remainder theorem If f(x) is a polynomial, then f(c) is the remainder of f(x) divided by (x – c)

Slope-intercept formula y = mx + b

Standard form ax + by = c

Sum of cubes formula (x3 + y3) = (x + y)(x2 – xy + y2)

TERMS – GEOMETRY

30-60-90 triangle A right triangle with one 30° angle and one 60° angle; sides measure x, x 3 , and 2x

45-45-90 triangle An isosceles right triangle; sides measure x and x 2

Apothem The distance from the center of a regular polygon to the middle of a side

Chord A line segment whose two endpoints lie on the circle

Circle All points equidistant from one center point (in two dimensions)

Cone A pyramid with a circular base

Congruent Having the same size and shape

Cylinder A circular prism

Midpoint The point on a line segment that is equidistant from both endpoints

Parallel lines Lines in the same plane that never intersect

Parallelogram A quadrilateral with two pairs of parallel sides

Perpendicular lines Lines that intersect at right angles

Prism Two parallel and congruent bases and the space between these two bases

Pyramid Has one base; its sides rise up from the base and meet at a vertex

Pythagorean triple Any three natural numbers that satisfy the Pythagorean theorem


Quadrilateral A four-sided polygon

Rectangle A parallelogram with four right angles

Rhombus A parallelogram with four congruent sides

Secant A line that intersects a circle in two points

Slope A line’s ratio of vertical to horizontal change

Sphere All points equidistant from one center point (in three dimensions)

Square A quadrilateral with equal sides and all right angles; is both a rectangle and a rhombus

Tangent A line that intersects a circle at only one point

Transversal A line that intersects two parallel lines

Trapezoid A quadrilateral with one pair of parallel sides

Triangle A three-sided polygon

Vertex Point of intersection of the sides of a pyramid or cone

FORMULAS AND THEOREMS – GEOMETRY

AA similarity theorem If two triangles exist such that two pairs of corresponding angles are congruent, then the triangles are similar

Chord-Chord Power Theorem Two intersecting chords form four line segments such that the product of one chord’s line segments equals the product of the other chord’s line segments

Distance formula In two dimensions: d = 212

212 )y-y(+)x-x( ;

in three dimensions: d = 212

212

212 )z-z(+)y-y(+)x-x(

Hero(n)’s formula A = c)-b)(s-a)(s-s(s ; s =

2c+b+a

Pythagorean theorem a2 + b2 = c2, where a and b are lengths of the two legs of a right triangle and c is the length of the hypotenuse

SAS similarity theorem If two triangles exist such that two pairs of corresponding side lengths form a constant ratio and the angles included between those sides are congruent, then the two triangles are similar

Secant-Secant Power Theorem The product of the lengths of one secant and its external part is equal to the product of the lengths of the other secant and its external part

Secant-Tangent Power Theorem The product of the lengths of the secant and its external part is equal to the square of the length of the tangent

SSS similarity theorem If two triangles exist such that all three pairs of corresponding side lengths form a constant ratio, then the two triangles are similar


TERMS – TRIGONOMETRY

Amplitude Half the distance between the maximum and minimum values of a cyclical wave function

Cosecant (csc) In a right triangle, the ratio of the length of the hypotenuse to that of the side opposite to the angle in question; reciprocal of sine

Cosine (cos) In a right triangle, the ratio of the length of the adjacent side to that of the hypotenuse

Cotangent (cot) In a right triangle, the ratio of the length of the adjacent side to that of the side opposite to the angle in question; reciprocal of tangent

Horizontal shift Sliding a graph along the x-axis

Inverse trigonometric function A function that “undoes” a trigonometric function

Law of cosines In a triangle, a way to find the length of an unknown side; c2 = a2 + b2 – 2ab(cosC)

Law of sines In a triangle, the ratio of the sine of each angle to its opposite side is

the same for all 3 angles; a

Asin = b

Bsin = cCsin

Period The interval over which a function repeats; all trigonometric functions are periodic

Reference angle The measure of the angle to the nearest x-axis; always between 0 and 90 degrees

Secant (sec) In a right triangle, the ratio of the length of the hypotenuse to that of the adjacent side; reciprocal of cosine

Sine (sin) In a right triangle, the ratio of the length of the opposite side to that of the hypotenuse

Tangent (tan) In a right triangle, the ratio of the length of the opposite side to that of the length of the adjacent side

Trigonometric identities Formulas that transform certain trigonometric expressions into other trigonometric expressions

Vertical shift Sliding a graph along the y-axis

TERMS – CALCULUS

Absolute maximum The maximum y-value a function attains; also known as global maximum

Absolute minimum The minimum y-value a function attains; also known as global minimum

Acceleration See second derivative

Antiderivative A possible function that has a known derivative

Concavity The direction that a curve is facing; found by taking the second derivative; a U shape is concave up; an upside-down U shape is concave down


Continuity A function is continuous at c if the limit as x approaches c equals f(c)

Critical point All the x-values at which a function’s derivative equals 0 or is undefined

Definite integral An integral with limits or bounds of integration; produces one value

Derivative The rate of change of the dependent variable with respect to the independent variable; instantaneous slope

Differentiation The process of taking a derivative

Displacement The distance a point is from its starting point; its first derivative is velocity; its second derivative is acceleration

First derivative The (instantaneous) slope of a function at a specific point; also known as velocity when the function is distance

First derivative test Used to determine if a relative max or min exists at a critical point; compare the first derivative of the function just before and just after the point; if the derivative changes from positive to negative, a min exists; if the derivative changes from negative to positive, a max exists; if it doesn’t change, no extreme is present at that point

Global maximum See absolute maximum

Global minimum See absolute minimum

Indefinite integral An unbounded integral; produces a set of possible functions

Integral See antiderivative

Integration The process of finding an antiderivative

L’Hopital’s rule If a limit is indeterminate, take the derivative of its numerator and its denominator, and re-evaluate the limit

Limit The y-value that a function approaches when getting arbitrarily and infinitely close to a given x-value

Local maximum See relative maximum

Local minimum See relative minimum

Point of inflection An x-value at which a function’s second derivative equals zero; marks a change in a graph’s concavity

Rate of change The rate at which one variable changes with respect to another variable under certain conditions

Related rate problem Problem that involves rates of change

Relative maximum A point whose y-value is greater than those of the surrounding points; located at the very peak of a graphical “hill”; also known as local maximum

Relative minimum A point whose y-value is less than those of the surrounding points; located at the very bottom of a graphical “valley”; also known as local minimum

Second derivative The instantaneous slope of the first derivative at a specific point; if positive, the graph is concave up; if negative, the graph is concave down; also known as acceleration when the function is distance


Second derivative test Used to determine if a relative max or min exists at a critical point; take the second derivative of a function and plug in a critical point; if the result is positive, a relative min exists at that point; if the result is negative, a relative max exists; if the result is zero, no extreme is present at that point

Tangent line A line that intersects a graph at only one point; the slope of this line at a specific point in the graph is equal to the derivative of the function at that point

Velocity See first derivative


POWER TABLE

Geometry: Shapes And Figures

Shape 2-D or 3-D? Area Formula20 Volume Formula Other Notes

Circle 2-D πr2 N/A 2-D set of all points a

certain distance (r) from a central point

Cone 3-D πr2 + πr 22 h+r 31

πr2h Pyramid with circular base

Cylinder 3-D 2πr2 + 2πrh πr2h Prism with circular base

Parallelogram 2-D bh N/A Has two sets of parallel sides

Pyramid 3-D (area of base) + (area of sides)

31

(area of the

base)(height)

Figure with one base; sides rise from base and

meet at a vertex

Rectangle 2-D Lw N/A Parallelogram with four right angles

Rhombus 2-D 21dd21

N/A Parallelogram with four congruent sides

Sphere 3-D 4πr2

34

πr3 3-D set of all points a

certain distance (r) from a central point

Square 2-D s2 N/A

Parallelogram with four right angles and four

congruent sides

Trapezoid 2-D )h)(b+b)(21

( 21 N/A Has one set of parallel sides

20 Area formulas given for 3-D shapes are surface area formulas.


POWER STRATEGIES

Beating the USAD Math Test Time management

Use a silent timer during practice and competition Divide the problems into sets of five

For each set of five, find one question that you’re fairly certain you can get right Attempt another question in the set that doesn’t seem too hard or long Unless you see another problem that you definitely know how to do, move on to the

next set Consider saving all trig identity questions to the end, since these tend to take the

longest After you have reached the end, go back and try the other questions This method allows you to find all the easy questions on the first pass

Do not spend too much time on any one question On average, each question should take less than a minute If you’ve spent more than two minutes on a question and are not close to having the

answer, move on At five minutes remaining, stop working on your current problem and guess on all of the

ones you have left blank This way, you at least have a chance of getting a few more points If you still have time left over after guessing, work on the one you just stopped

Learning the content Math is unique: it requires repetition of problem-solving skills, not memorization

Take practice tests often Then, ask a math teacher, coach, or fellow decathlete to teach you how to solve the

problems that you don’t understand Calculator use

Be familiar with all of the functions on your calculator Knowing where to find the most useful keys will save you time

Practice good calculator syntax Calculators interpret your input very strictly

Use parentheses to avoid miscalculations with fractions or exponents When using trig functions, make sure your calculator is in degree mode when working with

degrees and in radian mode when working with radians USAD’s calculator policy stipulates that all Decathletes must clear their calculators’ memory

before the start of the math test Having programs on your calculator, therefore, won’t be of much help

What to do when you don’t know the content Often, you may be able to plug the answer choices into the problem

Example: On trigonometric identities, you can choose random angles to substitute and check which answer choice matches the question

Before you begin a test, pick your favorite guessing letter


Use the same letter every time you guess without eliminating choices first21 Goals

Have a realistic number of questions that you want to get right An Algebra I student cannot realistically expect to achieve 35/35 Recently, even Calculus BC students have not been able to achieve 35/35

21 My team always chose D. – Dean

Shorter Selections Power Guide | 76

ABOUT THE AUTHOR Julia Ma grew up in Utah, where the snow is great and the mountains are mysterious. After being an alternate member on her first high school’s AD team, she loved AD so much that she started a new team when she transferred high schools. She recruited her friends onto the team, and they renamed the competition to “Akideki.”

Julia attended Caltech and recently graduated with a BS in Electrical Engineering. While at Caltech, she did research in various areas, such as computer graphics, robotics, insect vision, Albert Einstein’s history, and video-conferencing technology. She is always pleasantly surprised when she meets another Techer who used DemiDec materials and gloats that they probably took the math tests she wrote.

In her spare time, Julia likes to create music and art, collect computers (she now has three, each with a different operating system), and make homemade strawberry limeade. Take a box of strawberries and the juice from 3-5 limes (depending on how lime-y you like your drinks), put them in a blender with a cup of sugar, add ice to fill the rest of the blender, blend until smooth, go outside into the sunshine, sprawl onto the grass, and enjoy with friends.

Vital Stats: Competed with Alta High School as an honors decathlete in 2001-2002 Joined DemiDec in May 2002


ABOUT THE AUTHOR Steven Zhu joined the Frisco High School AcDec team in 2003. In his freshman year, he missed having a competing spot on the team by 0.3%. Undeterred, he eventually won the Texas individual state championship his senior year.

Steven was recruited to write the Math Power Guide in 2007 after being the only decathlete in the nation to break 900 points on the state math test. He currently studies economics, computer science, and Chinese at Harvard University, where he serves on the board of a student investment club, programs for the daily student newspaper, grades economics tests, and competes in ballroom dance. While writing this Power Guide, Steven was working his second summer as an intern at the Federal Reserve Bank of Dallas.

Vital Stats: Competed with Frisco High School as an honors decathlete at

regionals and at the Texas medium school state competition In 2006, team placed 2nd at regionals and 5th at state; individual

scores of 8355 and 8010, respectively In 2007, team placed 1st at regionals; individual score of 8509 In 2007, team placed 2nd at state; individually had the highest score in all divisions with 8823 Decathlon philosophy in a phrase: “Eat duplicate flashcards; make the knowledge yours” Joined DemiDec in March 2007


ABOUT THE CONTRIBUTOR Michael Nagle did not submit his author’s bio in time. However, we do know a few things about him. He helped lead the North Hollywood Academic Decathlon team to its first berth at the California state competition in 2001, under then-new coach Altair Maine. At MIT, he majored in mathematics and graduated in 2005. Vital Stats:

Competed with North Hollywood High School at the California state competition in 2001, finishing in eighth place

Team took fourth at the 2001 Los Angeles Unified School District Regional Competition Decathlon philosophy in a phrase: "[Decathlon] is like dreaming open-eyed"22 Joined DemiDec in 2005

22 Well, the original quote was about MIT…


ABOUT THE EDITOR Sophy Lee recently found out that her friend’s bearded dragons do not (usually) bite.

Over the years, Sophy has resigned herself to her stunning bad luck with aquatic animals. Her first pet, a goldfish that she won from a coloring contest, promptly died after she fed it white bread. The stuff is apparently as bad for fish as it is for humans.

Sophy’s second pet, a yellow fish named Shrimp, died on New Year’s Eve from a tragic fungal infection.

Next came Bonnie and Clyde—a brown and a red beta fish, respectively—that she raised with her Decathlon teammate, Edie. Clyde died after spending a week in the Pearland High School Acadec classroom and Bonnie, heartbroken, died soon afterwards.

This series of unfortunate events has convinced Sophy that Harvard University’s policy against pets is probably best for both her and the world’s fish population. As she heads into her second year of college, she plans to stay far away from laboratory animals and the campus’s possessed squirrels. You have a better chance of finding her muttering Russian in the library’s Language Resource Center, avoiding slushy snow on her way to class, or singing radio songs in an unmarked van on her way to a Mock Trial tournament.

If you have any suggestions about how Sophy can keep her pets alive, feel free to email her at [email protected].

Vital Stats: Competed with Pearland High School at the Texas Region V and Texas State competitions in 2007;

competed at Region, State, and Nationals in 2008 Team placed thirteenth at State in 2007; individual scores of 7,741 and 7,542 Team placed third at Nationals in 2008; individual scores of 9041, 9007, and 9304 Decathlon philosophy in a phrase: “No regrets” Joined DemiDec in June 2007


ABOUT THE EDITOR/POWER ALPACA Dean Schaffer believes that in his former life, he was either an owl (wise and nocturnal), a lolcat (prone to nonsensical utterances), or a Microsoft Word spellchecker (compulsive but vulnerable to glitches). In this life, he attends Stanford University, majors in American Studies, minors in Classics, and doesn’t really know what he wants to do when he grows up—something he constantly hopes he’ll never have to do.

Since joining DemiDec to write the Renaissance Music Power Guide, Dean has taken turns making the Power Guide more powerful, the flashcard a lot flashier, and the Cram Kit a bit…crammier? This season marks Dean’s fifth with DemiDec, and his lengthy tenure has, thus far, given him a glimpse of the ineffable quirks of the English language and, more notably, of the ineffable cuteness of the three puppies which inhabit DemiDec HQ (and are probably the single biggest productivity drain on DemiDec Dan).

In his spare time, Dean ponders whether he’ll ever be able to handle the luxury of spare time; luckily, he avoids this metaphysical quandary altogether by choosing not to affiliate himself with relaxation of any form. Instead, he occupies himself with songwriting, playing guitar, and parallel structure-ing. When he isn’t doing those things, he’s considering the merits of democratic elections, oligarchic disinterestedness, and delicious gouda cheese. Vital Stats: Competed with Taft High School in Los Angeles, California In 2005, team placed first at LA regionals and fifth at CA state with individual scores of 8792 and

8887, respectively In 2006, team placed first at LA regionals, CA state, and nationals with individual scores of 9121,

8903, and 8962, respectively Decathlon philosophy in a phrase: “Get back to work!” Joined DemiDec in April 2005


ABOUT THE BETA TESTERS Adriana Zamora ([email protected]) is a senior at Earl Warren High School and will be in Academic Decathlon for her third year. When Adriana doesn’t have her nose stuck in her study binder, she is with her twin sister, enjoying the quality time they have left before they possibly split up for college. She has a bittersweet relationship with procrastination and public speaking, which tends to be more bitter than sweet. She enjoys playing soccer, singing along to every song on the radio, and sleeping because she is usually deprived during the school year.

Quinn Campbell ([email protected]) spends most of his time trying to cram (seemingly) millions of facts into his head. Quinn then spends what little free time he has left learning about all sorts of other subjects that will never be tested by an AP or AcDec test. Quinn particularly enjoys economics, psychology, and space.

Erika Tinley ([email protected]) is entering her second year at Sonoran Science Academy and will be a senior. She hopes to win more shiny medals during her second year in Decathlon. She spends most of her time discreetly telling people that her coach was on "Jeopardy!" When she does have time, she uses it studying Decathlon and trying to graduate with credits galore. She also reads, shoots archery, has a motorcycle, and hopes to own a Harley.

Jane Huang ([email protected]) is entering her fourth year of Academic Decathlon at Walter Payton College Prep in Chicago. When not memorizing the names of terribly obscure musical instruments and other such minutiae for Acadec, she swims, plays the piano and the viola, competes on math team, and searches for random other details to insert in her unabashedly short-ish bio.

Anthony Sam Wu, also known by various monikers to different people (Tony, Panda-chan, "Anthany,” et al.), is a scholastic competing for Mark Keppel High, a school of sorts based in California. The photo shown explains a lot as to why Anthony is so strange (playing "Duke Nukem II" at age three? Really now).

Lawrence Lan doesn’t usually write about himself in the third person. What he does do on a usual basis is sleep—anytime, anyplace. When he is not sleeping irregularly, Lawrence finds satisfaction in good music, freeze-dried mango pieces, and The Office. A graduate and ex-Decathlete from Palos Verdes Peninsula High School in southern California, Lawrence currently attends Cornell University in Snowyville, New York—known by the locals as Ithaca.

Fermi Ma ([email protected]; not pictured) will be starting his junior year at Northside College Prep when September rolls around. He has been an active participant in his high school's Academic Decathlon Team and Math Team for the past two years. In his spare time, Fermi enjoys playing basketball, running, and solving math problems.


Miandra Ellis ([email protected]; not pictured) has been in Academic Decathlon for the past two years and this year will be her last. Miandra has learned only one thing from two years in the program: there is way more to learn out there.

Other beta testers who reviewed this Power Guide: Hillary Lam Benjamin Ferell Shiv Pande

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