Sequences, Series, and Probability. Infinite Sequences and Summation Notation.

UNIT 10Sequences, Series, and Probability

LESSON 10.1Infinite Sequences and Summation Notation

Lesson Essential Question (LEQ)

What is an infinite sequence and how do we use summation notation?

Infinite Sequence

An infinite sequence is a function whose domain is the set of positive integers.

Notation:

Hint: It’s the natural numbers from 1 to .

Examples:

Find the first 3 terms of the sequence, then find the 10th and 15th terms of the sequence.

1)

2)

Recursive Sequences

A recursive sequence is when you are given the first term of the sequence , and each term after is acquired by using the previous term.

Ex: Find the first 4 terms and the nth term for the following infinite sequence defined recursively:

Examples:

Find the first 5 terms of the recursively defined infinite sequence.

Homework:

Pages 735 – 736 #’s 1 – 13 odds, 21, 24, 25

Bell Work:

Find the first four terms and the 15th term for the sequence below:

1)

Find the first four terms and the nth term for the recursive sequence below:

2)

Summation Notation

Sometimes it is necessary to find the sum of many terms of an infinite sequence.

This represents the sum of the first n terms.

Examples:

Find the sum:

3)

4)

Sum of Constants

a)

b)

Examples:

Find the sum:

5)

6)

More Sum Theorems:

for all real numbers c

Homework:

Pages 735 – 736 #’s 22, 26, 27, 33 – 45 odds, 51, 54, 57, 58

Bell Work:

Looking at the following sequence:

7, 16, 25, 34, 43, 52…

What would be the 10th term in this sequence?

What would be the nth term in this sequence?

Find the sum of the 35th through 40th terms.

ARITHMETIC SEQUENCES

Lesson 10.2

OHHHH YEAH!!!

Arithmetic Sequences

A sequence is considered to be arithmetic if there exists a value d such that d can be added to the previous term to find the next term.

To put it simply:

The value d is called the common difference.

Find the nth term…

When looking at a sequence, we can easily determine the nth term by finding the common difference.

Examples:

1) The first term of an arithmetic sequence is 20, and the sixth term is -10. Find the nth term and then find the 20th term.

2) If the 4th term of a sequence is 5 and the 9th term is 20, find the nth term and then find the 50th term.

Homework:

Pages 742 – 743#’s 3 – 17 odds

Bell Work:

For the arithmetic sequences below, find the 5th, 25th, 100th, and nth terms.

1) -12, -7, -2, 3…

2) 12.5, 9.2, 5.9, 2.6…

3) If the 8th term of a sequence is 32 and the 12th term is -16, find the nth term and then find the 50th term.

Partial Sums

Suppose is an arithmetic sequence with a common difference d, then we can find the partial sum by using the first n terms.

or

Examples:

Find the sum of every even integer from 2 through 300.

Find the sum of every other odd integer from 1 through 513.

Inserting Arithmetic Means

Its exactly like it sounds! We are finding averages between values.

We are trying to find values that are equidistant from each other, so we have to find the common difference!!!

Examples:

Ex: Insert three arithmetic means between 2 and 10.

Ex: Insert seven arithmetic means between 18 and 24.

Expressing a Sum in Summation Notation!!!

Ex:

Ex:

Homework:

Page 743 #’s 27 – 45 odds

Bell Work:

Consider the following recursive sequence:

1) If the first term in the sequence is given as 10, what will happen to the terms of the sequence as k gets much larger?

2) What would happen if the first term was 10,000?

Quiz Coming Up!!!!!

Infinite Sequences Summation Notation Arithmetic Sequences

Bell Work:

1) Every day after soccer practice, Bobby loses 15% of the sodium in his body through perspiration . With the dinner he eats after practice, he intakes 40 mg of sodium. Write a recursive sequence to show the amount of sodium in Bobby’s system after any given day.

2) If Bobby initially has 400 mg of sodium in his system, how long will it take to drop below 350 mg?

3) If Bobby wants to maintain a level of 380 mg of sodium in his system, how many mg of sodium should he consume after practice each night?

Class Work/Home Work:

Pages 735 – 737 #’s 14, 24, 42, 52 (review 57 and 58)

Pages 742 – 744 #’s 6, 8, 12, 16, 18, 22, 26, 32 – 44 evens

We will review these on Monday, and take a quiz on Tuesday!!!

Bell Work:

What would be the 8th term in this sequence?

GEOMETRIC SEQUENCES

Lesson 10.3

Can I get a WHAT WHAT?!?!?!?!

Geometric Sequences

A sequence is geometric if there exists a real number r such that the next term in a sequence can be found by multiplying the previous term by r.

, this is called the common ratio.

Examples:

Given the following geometric sequence, find the common ratio:

Ex:

Ex: 200, 160, 128…

Find the nth term…

This will allow you to find any term of a geometric sequence.

This can also be used to obtain the common ratio r or the first term

Examples:

Ex1: A geometric sequence has a first term of 8 and a common ratio of -1/2. Find the first five terms and the nth term of the sequence.

Ex2: The 3rd term of a geometric sequence is 5 and the 6th term is -40. Find the nth term and the 11th term of the sequence.

Examples:

Ex: If the third term in a geometric sequence is and the 8th term is , find the nth term and the 5th term of the sequence.

Homework:

Page 751 #’s 3 – 19 odds only

Bell Work:

If the third term in a geometric sequence is and the seventh term in the sequence is -270, then find the nth term and the 10th term.

Partial Sums!!!! Yeah boiiiiiiiiii!

The partial sum of a geometric sequence can be found by:

, where r ≠ 1.

Remember, this is different from an arithmetic sequence!

Example:

Suppose Mr. Kelsey decides to save up to buy a castle. He sets aside 1 cent on the first day, 2 cents on the second, 4 on the third, 8 on the fourth, and so on.

A) How much money will he have to set aside on the 18th day?

B) What is the total amount of money he will have saved up after 25 days???

Sum of an Infinite Geometric Sequence!!!!!

If -1 < r < 1, then an infinite geometric series has the sum of:

Ex: Lets find the sum of the following series:

We can only find sums of infinite sequences if the sequence if the common ratio is between 1 and -1!!!!!

Examples:

Find the sum of the following infinite geometric sequences:

Ex:

Ex:

Ex:

Homework:

Pages 751 – 753 #’s 21 – 25 odds, 29 – 35 odds, 45 – 55 odds and 58

Bell Work:

The yearly depreciation of a certain machine is 20% of its value at the beginning of the year. If the original cost of the machine is $400,000, then what is its value after 15 years?

Find the sum of the following infinite geometric series:

LESSON 10.5The Binomial Theorem

Before we get there…

Factorial!!!!!!!!!!!!

6! is read as 6 factorial.

6! = 6 x 5 x 4 x 3 x 2 x 1 = 720

4! = 4 x 3 x 2 x 1 = 24

8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40320

Combinations:

In order to determine binomial coefficients, we need to know how to find a combination.

C(n,k) =

Ex: Lets find C(5,3)

Examples:

Find the following combinations:

Ex: C(8,4)

Ex: C(10,3)

Ex:

The Binomial Theorem

What does it do?

It allows us to expand any binomial raised to the nth power!

We know that (a + b)² = a² + 2ab + b², but what about expanding (a + b) to the 10th power???

Here we go…

Warning: What you are about to see can be confusing and downright insane, viewer discretion is advised.

Turn to page 765 in your textbooks and look at the top orange table.

This is…The Binomial Theorem.

Let’s see how it works…

Let’s expand

Okay, that was nice, but how about this?????

Examples:

Try a few on your own!

Ex:

Ex:

Homework:

Page 769 #’s 5, 11, 19, 23, 25

Bell Work:

Simplify:

1) C(9,5)

2)

3) Expand:

Expanding Binomials Practice:

Expand:

1)

2)

Pascal’s Triangle

Get ready to be amazed at Pascal’s Triangle…

It used to be called Kelsey’s Triangle, but I am currently in a legal battle over naming rights.

Homework:

Page 769 #’s 26, 28, 31, 33, 35, 37, 39

Bell Work:

Use Pascal’s Triangle to expand .

Now use this to expand

Remember…

When expanding binomials, you will always have (n + 1) terms.

will have 10 terms

You can use the Binomial Theorem discussed in class, or Pascal’s Triangle. Both will work!!!

Finding a Specific Term:

To find a specific term, we use the following:

The (k + 1) term = for some binomial .

Example: Find the fifth term of

ClassWork/Homework:

Page 769 #’s 31 – 43 odds

Bell Work:

We have a quiz Tuesday/Wednesday, so try the following review problems:

Pages 751 – 753 #’s 10, 20, 28, 34, 36, 48, 50, 53

Page 769 #’s 10, 12, 24, 28, 34, 36, 40

PREVIEW OF BASIC CALCULUSRelax, it will be fun.

DIFFERENTIATION RULES

Derivatives of Polynomials and Exponential Functions

Brief Intro into Derivatives

Tangent Lines Slope Maximum/Minimum Distance/Velocity/Acceleration Zeroes Limits

Notation:

For a given function , its derivative is given as

It goes the same for any function:

The derivative of g(x) is g’(x).

Power Rule

For any real number n, where , the derivative of the function is given as:

Constant Rule

The Derivative of a Constant is ZERO!

If f(x) = c, where c is a constant, then f’(x) = 0.

The Sum/Difference Rule:

If f and g are both differentiable functions, then:

If h(x) = f(x) + g(x), then h’(x) = f’(x) + g’(x)

and

If h(x) = f(x) - g(x), then h’(x) = f’(x) - g’(x)

Finding the Derivative of Polynomial Expressions:

Find the derivative of the following expressions:

Ex:

Ex:

Ex:

Bell Work:

The maximum and minimum x values for a function can be found by finding the zeroes for the derivative of the original function.

Find the exact values of the maximums and minimums for the following function by using the derivative:

Example:

Use the derivative of the following function to find the exact values of the maximums and minimums of the original function.

Second/Third/Fourth…Derivatives

Given the function:

Find the first derivative.

Find the second derivative.

Find the third derivative.

Derivative of the Natural Exponential Function

If then

Product Rule

If f and g are both differentiable functions, then:

If h(x) = f(x)· g(x),

then h’(x) = f(x)· g’(x) + g(x)· f’(x)

Example:

Find the derivatives of the following function:

Ex:

Quotient Rule

If , then

Example: Find the derivative of:

Bell Work:

Find the first derivatives of the following functions:

1)

2)

3)

DERIVATIVES OF TRIGONOMETRIC

FUNCTIONSYeah, that’s right, this is

happening.

Challenge Question:

What is the first derivative of f(x) = sin(x)?

Sine and Cosine

If then .

If then .

Ex: Find the first derivative of and

Tangent

Let’s derive the first derivative of tan(x).

The rest of the riff raff…

If then .

If then .

If then .

If then .

Bell Work:

1) Prove that the first derivative of

2) Prove that the first derivative of

Examples:

Differentiate the following:

Challenge Problem:

Find the 39th derivative of f(x) = cos(x).

CHAIN RULE FOR DIFFERENTIATION

Chain Rule

When finding the derivative of a function that contains a composition, we must use the chain rule.

How do we use it?

Examples: Find the first derivative of the following functions:

Bell Work:

Differentiate the following:

1)

2)

3)

Logarithms!!!

If , then

Example: Differentiate

NATURAL LOGARITHMS!!!

If f(x) = ln(x), then .

Example: Differentiate

Example: Differentiate

Combining the Chain Rule…

Sometimes it is necessary that in order to find a derivative, you must use the chain rule with another rule (usually the product or quotient rule).

Ex:

Homework:

Differentiate:

Find the 1,231st derivative of f(x) = sin(x)

Bell Work:

Find the derivative of

Find the derivative of

Challenge Problem:

Differentiate

If you don’t enjoy this problem, then you have serious issues that need to be worked out.

Double Bonus Challenge Problem!!!!!!!!!!!!

There really is no bonus, but it’s a Thursday double day DOUBLE CHALLENGE PROBLEM YEAH!!!!!!!!!!!!!!!

Find the derivative of

TRIPLE BONUS CHALLENGE PROBLEM!!!!!!!!!!!!!!!!

OMG! It’s a Thursday Triple Day TRIPLE BONUS CHALLENGE PROBLEM!!!!!!

Find the derivative of

QUADRUPLE BONUS CHALL…OKAY, NOT REALLY.

No homework, review tomorrow and Monday. Test will be on Tuesday or Wednesday next week.

Arithmetic Sequences Geometric Sequences Binomial Expansion (Binomial Theorem or

Pascal’s Triangle) Derivatives of Polynomial Functions

Tomorrow’s Assignment

Pages 799 – 800

#’s 1, 8, 10, 17, 25 – 30, 39, 50 – 52

These are only on Sequences and Binomial Expansions, not derivatives, they will come on Monday!