Flashcards AP Calc Final Exam Set 1

8/3/2019 Flashcards AP Calc Final Exam Set 1

1/18

What are the key features one should

locate on a derivative table?

For any given wire, the fundamental frequency of that wire is the

following function:

1

2

Ty

rl d

=

r: the radius of the wired: the density of the wire

T: the tension (force) holding the wire

l: the length of the wire

Find dy/drassuming l, d, and Tare constants.

What characteristics make a function EVE

or ODD?

What must you always remember to do

when taking the anti-derivative, and your

solution contains natural log?

How do you find the y value of a hole

Show the process for findingdxdt

whendydt

=2, x = 3, and 2 36x y = .

What do you need to be careful about when

solving for f (x) in the following equation?

( )32( ) ( ) 12xf x x f x =

Use the following information to determine an

approximation for f(5.15)

f(5)=4.5

f (5) = 1.1

(5.15, ? )


2/18

Separate the constants from the variable:

1

2

Ty r

l d

=

So

( )2 2or2 2

dy T T r

dr l d l dr

=

- Any location where f (x) = 0

-Any location where f (x) =

Use the tangent line:

y = 1.1(x 5) + 4.5

f(5.15) 1.1(5.15 5) + 4.5 = 4.665

f(5.15) 4.665

Use a limit!

Example:

23

23

3 1lim , this is a hole!

9 6

3lim , this is a vertical asymptote.

9

x

x

x

x

x

x

=

=

Use an Absolute Value!

Example:

2

2'( )

4

( ) ln 2 4

xf x

x

f x x C

=

= +

Distribute the negative!You should get the following:

( ) ( )22

33 ( ) '( )( ) '( ) 2 ( ) 0x f x f xf x xf x x f x + =

NOT:

( ) ( )22

33 ( ) '( )( ) '( ) 2 ( ) 0x f x f xf x xf x x f x ++ =

Rewrite:

x(t)2y(t) = 36

2x(t)x (t) y(t) + x(t)2 y (t) = 0

plug in your values: 2(3)x(t)(4)+(3)2(2) = 0

24x (t)=-18 so: x (t) = -0.75

EVEN:

f(x) = f(-x) OR y-axis symmetry

ODD:

- f(x) = f(-x), OR 180o rotational

symmetry to the origin.

No, no, no!

Note: when x = 3, y = 4


3/18

Give a simple definition for a function be

continuous at a given point a.

What is the average value of f(x) on the

interval [0,4], given

f(x) = -3x2 + 4x?

Is it possible for a function to be continuo

yet nothave a defined derivative at a poi

a?

State thesymmetric difference

quotientform of the derivative.

State the asymmetric difference

quotientform of the derivative.

Given c (t) = -0.125 (c(t) 750 ),

and c (0) = 900, how do you

find an approximation

for c(1) using ten steps of 0.1?

Under what condition can you apply the

Mean Value Theorem?

or a piecewise function to be differentiable

at a point a, what

two things have to be true?


4/18

( )4

2

03 4

4 0

x x dx +=

( ) ( )3 2 3 2(4) 2(4) (0) 2(0)4

+ +=

64 32 328

4 4

+ = =

lim ( ) ( )x a

f x f a

=

You need to check the following two things:

1) The function needs to be continuous at point a

or: lim ( ) ( )x a

f x f a

=

2) The f (x) needs to exist at point a.

or: lim '( ) '( )x a

f x f a

=

0

( ) ( )lim '( )h

f x h f xf x

h+ =

0

( ) ( )lim '( )

2hf x h f x h

f xh

+ =

When the function is both

continuous and differentiable on

the interval given.

1) Enter c (t) into Y1

2) Regular screen: enter 900, press enter3) Enter: Y1(ANS) ( 0.1) + ANS

Use 2nd ENTER command to repeat ten times

Answer: approximately 882.27

YES!

If the function has a sharp point or a

cusp, the function would becontinuous at the cusp, but the

derivative would notexist.


5/18

The rate of consumption of cheese wiz is give

by s(t) = Cekt.

If the consumption doubles every 10 years,

with an initial consumption of 10,000 tons pe

year, find values for C and k.

3 2( 6) ?x dx+ =

1) tan( ) ?x dx =

and

2) ( ) tan( ), ?dyf x xdx

= =

State all three log rules:

What is the quotient rule?

What does the Mean ValueTheorem mean?

What two rules should you remember wh

solving an expression?

When looking for the absolute

maximum orabsolute minimum of a

function on a given interval, what must

you remember to do?


6/18

Cant do it straight, so FOIL it out!

( )3 2 6 3( 6) 12 36x dx x x dx+ = + +

7 41 3 367x x x C + + +

ug in two data points into the given equation. These datapoints are ( 0, 10 ) and (10, 20):

10 = Cek(0), 10 = C

and

20 = 10 ek(10)

, 2 = e10k

, ln(2) = 10k,

k=ln(2)

10

You mustremember to check the

extremes / endpoints.

( )2

( )

( ) ,( )

'( ) ( ) ( ) '( )'( )

( )

g x

if f x h x

g x h x g x h xthen f x

h x

==

( )

1)log ( ) log ( ) log ( )

2)log ( ) log ( ) log

3)log log ( )

d d d

d d d

bd d

a b aba

a bb

a b a

+ = =

=

1) Get rid of negative exponents:

make into fractions.

2) f-a-c-t-o-r, f-a-c-t-o-r, f-a-c-t-o-r!

If the graph is differentiable, and you were to

draw a line connecting the endpoints, then

there is a point(s) somewhere on the curve

that has a tangent line that is parallel to the

secant line connecting the endpoints.

Note: the point you are looking for is

between the endpoints.

1) ln |cos( ) |x C +

and

2) 221 or sec ( )cos ( )x

x


7/18

What is the derivative of a sharp

point or a cusp?

What does symmetric to the origin mean?

What does symmetric to the y-axis mean?

When taking the anti-derivative of a

function with a power of -1, what musty

remember?

What is the difference between a

right-handed Riemann Sum and aleft handed Riemann Sum?

Construct a sequence for an integral

evaluated by using 6 right-handedrectangles of equal width on the

interval [a,b].

Construct a sequence for an integral

evaluated by using 6 left-handedrectangles of equal width on the

interval [a,b].

State two rules for solving differential

equations (solve for y that does not conta

dy/dx).

f(x) = (3x 2)2(x + 5)3.

Find all value where f (x) = 0


8/18

Symmetric to the Origin means an ODD functi

(180 degree rotational symmetry around the orig

Symmetric to the y-axis means an EVEN Funct

F(x) = f(-x)

Note: The two definitions above are assuming we ar

dealing with functions.

Undefined, nonexistent.

f (x) =2 (3x 2)1(3)(x + 5)3+(3x 2)2 (3)(x + 5)2

FACTOR!

0 = (3)(3x 2)(x + 5)2( 2(x + 5) + (3x 2) )

0 = (3)(3x 2)(x + 5)2( 2x + 10 + 3x 2) )0 = (3)(3x 2)(x + 5)2( 5x + 8 )

solutions: x = 2/3, -5, and -8/5

( )( ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( )w f a w f a w f a w f a w f a w f b+ + + + + + + + + +

note: Start one width past a and stop at b.

Right handed sum:

Left handed sum:

1) Both f(x) and f (x) must be on the same

side of the equation.

2) Only use the natural log if you have apowerof -1.

( )( ) ( ) ( 2 ) ( 3 ) ( 4 ) ( 5 )w f a f a w f a w f a w f a w f a w+ + + + + + + + + +

note: start at a, stop one width short ofb.

your anti-derivative ends up being natural log, remember

to use absolute value!

Example:

f xx

x

f x x C

' ( ) ,

( ) l n

=

= +

3

2 6

3

42 6

2

2

a b

6

b aw

=

6

b aw

=

a+w a+2w


9/18

What do you do/use to determine if a

the graph of a curve is or is

not a function?

What is thesymmetric difference quotient?

What is it used for?

What is a Reimann Sum?

Stupid Algebra:

What is wrong with the following:

x x2 9 3 =

Stupid Algebra:What is wrong with the following:

x

xx

2 88

=

Do the zeroes of the 2nd derivative

always locate points of inflection?

State 3 different forms for finding the area of the follow

triangle:

What equation gives you the average value

of a given function?

Example: What is the average value of

v(t) = 4t2 4t on the interval [-2,6]?

A

B

Ca

b

c


10/18

0

( ) ( )lim

2h

f x h f x h

h

+

It is used to find the derivative, or velocity,

of an function

Use the vertical line test

( )average value on [a,b]

b

af x dx

b a=

example:

( )6

2

24 4

6 2

t t dt average

=

Must cancel an x in both terms:

x

x x x

2 8 8= ,

you can test with numbers!

2 8

22 8

2

Cant take the square root like that! You can test usinnumbers:

2 9 2 3

5 1

2

Use one of the following:

1sin( )

21

sin( )21

sin( )2

ab C

ac B

cb A

NO!

You need to make a 2nd derivative table to determine

there is, in fact, a change in concavity.

f(2) is nota point of inflection, f(6) is

When you create a sum of a

series of rectangles or

trapezoids to find the area

under a curve.

f(x)

f (x)

2 6

0 0>0 >0


11/18

What conditions must be met so that the follow

function would be differentiable

on the interval [-10,10]?

4 , 2( )

, 2

x a xf x

bx x

+


12/18

f (x) = 0 at -4, 0, and 4:

Check table, points of inflectionare at x = 0 and x = 4 only

f (x) and f (x) must be continuous at x = 2:

or

16 + a = 2b and 4(2)3 = b,b =32

a = 2(32) 16 = 48

(2)(15)log (5) log

6

log (2) log (15) log (6)

c c

c c c A C B

= =

+ = +

( ) ( )tan tan'( ) lim2

e h e h

h o

e ef e

h

+

=1

2

2 2

1

2

=

( 4)

( 4) ( 4)

( 4)x

x

dt dt

dx

d t dt xdx

x

=

V = r2h

SA = 2r2 + 2rh

They would have the same solutions,

except f x d a

b

( ) would have the

opposite answer to f x d b

a

( )Example:

if f x d a

b

( ) =12, then f x d b

a

( ) =-12

22

1( 4)( 4)

xx

d t dtdx

=

0 0 0

-4 0 4f(x)

f (x) >0 >0 0


13/18

The following integral was written to findthe volume created by rotating region R

360 degrees around the line y = 3.

What is wrong with it?




The following integral was written to find

the volume created by rotating region R




360 degrees around the line x = 2.











The following integral was written to find

the volume created by rotating region R



R

y = 3

5.26

f(x) = 3(x-4)

-4

( ) ( )5.26 2 24

0 3 4 3x

dx

R

y = 3

5.26

f(x) = 3(x-4)

-4

( )( ) ( )( )25.26

4

03 3 4 3x dx

R

y = 3

5.26

f(x) = 3(x-4)-4

( )( ) ( )( )25.26 24

00 3 4 3x dx

R

y = 3

5.26

f(x) = 3(x-4)-4

( )( ) ( )( )25.26 24

03 3 4 3x dx

f(x) = 3(-x-4)

-4

R

x = 2

-5.26

( ) ( )( )0

4

5.262 2 0 3 4xx dx

f(x) = 3(-x-4)

-4

R

x = 2

-5.26

( ) ( )0

4

5.262 2 0 3 4xx dx

f(x) = 3(-x-4)

-4

R

x = 2

-5.26 ( ) ( )0

4

5.262 2 3 4xx dx

f(x) = 3(-x-4)

-4

R

x = 2

-5.26

( ) ( )( )0

4

5.26

2 2 0 3 4xx d


14/18

This integral is based upon the idea of finding the arecontained between two concentric circles. The

larger circle would have a radius of3 f(x).

It has to be the distance fromy = 3down to the function.

This integral is based upon the idea of finding the area

ontained between two concentric circles. You cannot findthat area by just subtracting the two radii.

(r1)2

- (r2)2 (r1 r2)2

This integral is based upon the idea of finding the are

contained between two concentric circles. The

larger circle would have a radius of3 f(x).

It has to be the distance fromy = 3down to the function, not zero.

his integral is based upon the idea of a series of cylinders.The lateral area of the cylinder is

Given by 2 rh. h is the height of the cylinder, which is

measured from the x-axis down to the function. You needgroup the function to make sure you are subtracting the

entire function from zero.

This integral is based upon the idea of a series of cylindThe lateral area of the cylinder is

Given by 2 rh. ris the distance from the center of rota

(x = 2) and the wall of your cylinder (x).You always measure length on a graph

from rightto left.

Nothing, this one is correct!

This integral is based upon the idea of a series of cylindThe lateral area of the cylinder is

Given by 2 rh. h is the height of the cylinder, which

measured from the x-axis down to the function. You ne

to group the function to make sure you are subtracting

entire functionfrom zero.

Nothing, this one is correct!

( )( ) ( )25.26 24

03 3 4 3x dx

( )( ) ( )25.26 24

03 3 4 3x dx

Note the extra the 3

( )( ) ( )25.26 24

03 3 4 3x dx

Note: The 0 is changed to 3

( ) ( )( )0

4

5.262 2 0 3 4xx dx

Change radius to ( 2 x )

( ) ( )( )0

4

5.262 2 0 3 4xx dx

Note: insert

( ) ( )( )0

4

5.262 2 0 3 4xx dx

Note: 0 was left out, and group with

Keep the two radii separate


15/18

There is a third common limit form of

the derivative. In other words, a form

that is not the asymmetric or

symmetric difference form. What is

this third form?

For the following:

a < b < c, ( ) 4 and ( ) 12,c a

a bf x dx f x dx= =

find 5 ( )c

bf x dx

What equation is used to find the length of a

continuous differentiable curve on the

interval [a.b]?

Under what conditions can you useLHopitals rule?

What is the difference between the

following two statements:

( )1

( ) cos( )xd

f x t dt dx

= and

( )3

1( ) cos( )

xdf x t dt

dx=

Complete the following:

20

1 cos( )lim ?x

x

x x =

+

Complete the following:

20

sin( )lim ?

x

xx+

=

If y = kx + 7 is tangent to the graph of

f(x) = 3x3 + cos(x), what two (2)

conditions must be true?


16/18

Change 12( ) 12 to ( )a b

b af x dx f x dx = =

-Answer:

16, so:

5 80

( )

5 ( ) ( )

c

b

c c

b b

f x dx

f x dx f x dx

=

= =

( ) ( )lim '( )x a

f x f a

x af a

=

Note: This is another form of a slope between

two points that are very close together. You

know these two points are close together b/c of

the limit xa

1) They must share a common slope:

k= 9x2 sin(x)

2) The must share a common point:

kx + 7 = 3x3 + cos(x)

( )1

( ) cos( ) cos( )xd

f x t dt dx

x= =

( )3

1

2 3( ) cos( ) 3 cos( )

xdf x t dt x

dxx= =

When you are taking a limit, and you

are getting either or00

Use LHopitals Rule

20

0

0

0

1

0

sin( )lim

cos( )lim

2

x

x

xx

xx

+

+

=

= =

Use LHopitals Rule

20

0

00

sin( ) 0 01 2 1

1 cos( )lim

lim

x

x

xx

x

x x

= =+

=+

( )2

1 '( )b

af x dx+

note: If you find the antiderivative, plug in x3

and 1,

then take the derivative, you get the extra 3x2

in front


17/18

If y = 4sin -1(3x2),

find dy/dx

(2 ) ?xcos x dx =

Show how to find the derivative of :

2log (5 )x = f(x)

What are the derivatives of:

1) f(x) = arcsin(x)

2) f(x) = arcos(x)


18/18

Integration by parts!

(2 )xcos x dxf(x) = x, f (x) = 1,

g(x) = 0.5sin(2x), g (x) = cos(2x)1 1

sin(2 ) sin(2 )2 2

1 1sin(2 ) cos(2 )2 4

x x x dx

x x x C

=

+ +

4

24

1 9

dy xdx x

=

1) 21

1 x

2) 21

1 x

Use logarithmic differentiation:( )2 5

( ) ln(2) ln(5 )

5'( ) ln(2)

5

1'( )ln(2)

f x x

f x x

f xx

f xx

==

=

=

Note: ln(2) is

a number,

therefore itsderivat8ive is

Flashcards AP Calc Final Exam Set 1

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