Sequences, Series, and Probability

Infinite Sequences and Summation Notation

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

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

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

 Looking at the following sequence:  7, 16, 25, 34, 43, 52…  What would be the 10 th term in this sequence?  What would be the nth term in this sequence?  Find the sum of the 35 th through 40 th terms.

Lesson 10.2 OHHHH YEAH!!!

 1) The first term of an arithmetic sequence is 20, and the sixth term is -10. Find the nth term and then find the 20 th term.  2) If the 4 th term of a sequence is 5 and the 9 th term is 20, find the nth term and then find the 50 th term.

 Pages 742 – 743#’s 3 – 17 odds

 For the arithmetic sequences below, find the 5 th, 25 th, 100 th, and nth terms.  1) -12, -7, -2, 3…  2) 12.5, 9.2, 5.9, 2.6…  3) If the 8 th term of a sequence is 32 and the 12 th term is -16, find the nth term and then find the 50 th term.

 Find the sum of every even integer from 2 through 300.  Find the sum of every other odd integer from 1 through 513.

 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!!!

 Ex: Insert three arithmetic means between 2 and 10.  Ex: Insert seven arithmetic means between 18 and 24.

 Page 743 #’s 27 – 45 odds

 Infinite Sequences  Summation Notation  Arithmetic Sequences

 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?

 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!!!

Lesson 10.3 Can I get a WHAT WHAT?!?!?!?!

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

 Page 751 #’s 3 – 19 odds only

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

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

The Binomial Theorem

 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

 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 10 th power???

 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.

 Page 769 #’s 5, 11, 19, 23, 25

 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.

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

 Page 769 #’s 31 – 43 odds

 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

Relax, it will be fun.

Derivatives of Polynomials and Exponential Functions

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

 The Derivative of a Constant is ZERO!  If f(x) = c, where c is a constant, then f’(x) = 0.

 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)

 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)

Yeah, that’s right, this is happening.

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

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

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

 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

 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!