Monday, Oct 26, 2015MAT 146 Test #3! NO CALCULATOR! Thursday (STV 229) CALCULATOR OK! Friday (STV...
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Transcript of 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!
Monday, Oct 26, 2015
MAT 146
Polynomial Approximators
Monday, Oct 26, 2015
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 146Monday, Oct 26, 2015
MAT 146Monday, Oct 26, 2015
MAT 146Monday, Oct 26, 2015
MAT 146
Polynomial Approximators
Monday, Oct 26, 2015
Goal: Generate polynomial functions to approximate other functions near particular values of x.
Create a third-degree polynomial approximator for
MAT 146Monday, Oct 26, 2015
Create a 3rd-degree polynomial approximator for
MAT 146
What is an Infinite Series?
Monday, Oct 26, 2015
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?
Monday, Oct 26, 2015
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
Monday, Oct 26, 2015
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
Monday, Oct 26, 2015
The harmonic series is the sum of all possible unit fractions.
MAT 146
Notable Series
Monday, Oct 26, 2015
A telescoping sum can be compressed into just a few terms.
MAT 146
Fact or Fiction?
Monday, Oct 26, 2015
MAT 146
Applications!
Monday, Oct 26, 2015
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
Monday, Oct 26, 2015
Details . . . details . . . details!
https://plus.maths.org/content/teacher-package-differential-equations