monday, oct 26, 2015mat 146 test #3! no calculator! thursday (stv 229) calculator ok! friday (stv...

Calculus II (MAT 146)Dr. Day Monday, Oct 26, 2015

Sequences and Series (Chapter 11)

WA Tasks 11.1 and 11.2 due tonight Differential Equations Review due Wed

10/28

Monday, Oct 26, 2015 MAT 146

Test #3!NO CALCULATOR! Thursday (STV

229)CALCULATOR OK! Friday (STV

219)

MAT 146

Some Sequence Calculations

① If an = 2n−1, list the first three terms of the sequence.

② The first five terms of a sequence bn are 1, 8, 27, 64,

and 125. Create a rule for the sequence, assuming this pattern continues.

③ For the sequence cn = (3n−2)/(n+3) :

i) List the first four terms.ii) Are the terms of c

n getting larger? Getting smaller? Explain.

iii) As n grows large, does cn have a limit? If yes, what is it? If

no, why not?

④ Repeat (3) for this sequence:

⑤ Give an example of L’Hôspital’s Rule in action.Monday, Oct 26, 2015

MAT 146

Some Sequence Calculations

① If an = 2n−1, list the first three terms of the

sequence: {1,3,5}

② The first five terms of a sequence bn are 1, 8, 27,

64, and 125. Create a rule for the sequence, assuming this pattern continues. b

n = n3

③ For the sequence cn = (3n−2)/(n+3) :

i) List the first four terms: 1/4 , 4/5 , 7/6 , 10/7

ii) Are the terms of cn getting larger? Getting

smaller? Explain.iii) As n grows large, does c

n have a limit? If yes,

what is it? If no, why not?

④ Repeat (3) for this sequence:

⑤ Give an example of L’Hôspital’s Rule in action.

Monday, Oct 26, 2015

MAT 146Monday, Oct 26, 2015

Sequence CharacteristicsConvergence/Divergence: As we look

at more and more terms in the sequence, do those terms have a limit?

Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence.

Boundedness: Are there values we can stipulate that describe the upper or lower limits of the sequence?

MAT 146

Why Study

Sequences and Seriesin Calc II?

Taylor Polynomials applet

Infinite Process Yet Finite Outcome . . . How Can That Be?

Transition to Proof

Re-Expression!


http://www.personal.psu.edu/dpl14/java/calculus/taylorseries.html

MAT 146

Polynomial Approximators


On of our goals this chapter is to generate polynomial functions that can be used to approximate other functions near particular values of x.

The polynomial we seek is of the following form:

MAT 146

Polynomial Approximators


Goal: Generate polynomial functions to approximate other functions near particular values of x.

Create a third-degree polynomial approximator for


Create a 3rd-degree polynomial approximator for

MAT 146

What is an Infinite Series?


We start with a sequence {an}, n going from 1 to ∞, and define {si} as shown.

The {si} are called partial sums. These partial sums themselves form a sequence.

An infinite series is the summation of an infinite number of terms of the sequence {an}.

MAT 146

What is an Infinite Series?


Our goal is to determine whether an infinite series converges or diverges. It must do one or the other.

If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges.

MAT 146

Notable Series


A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge?

MAT 146

Notable Series


The harmonic series is the sum of all possible unit fractions.

MAT 146

Notable Series


A telescoping sum can be compressed into just a few terms.

MAT 146

Fact or Fiction?


MAT 146

Applications!


Spreading a Rumor: Suppose that y represents the number of people that know a rumor at time t and that there are M people in the population. For these parameters, one model for the spread of the rumor is that “the rate at which the rumor is spread is proportional to the product of those who have heard the rumor and those who have not heard it.”

MAT 146

Application: Ice Growth


Details . . . details . . . details!

https://plus.maths.org/content/teacher-package-differential-equations



monday, oct 26, 2015mat 146 test #3! no calculator! thursday (stv 229) calculator ok! friday (stv...

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