Seating by Group Thursday, Nov 10 MAT 146

Calculus II (MAT 146) Dr. Day Thursday, November 10, 2016 Integration Applications Area Between Curves (6.1) Average Value of a Function (6.5) Volumes of Solids (6.2, 6.3) Created by Rotations Created using Cross Sections Arc Length of a Curve (8.1) Probability (8.5) Methods of Integration U-substitution (5.5) Integration by Parts (7.1) Trig Integrals (7.2) Trig Substitution (7.3) Partial-Fraction Decomposition (7.4) Putting it All Together: Strategies! (7.5) Improper Integrals (7.8) Differential Equations What is a differential equation? (9.1) Solving Differential Equations Visual: Slope Fields (9.2) Numerical: Euler’s Method (9.2) Analytical: Separation of Variables (9.3) Applications of Differential Equations Infinite Sequences & Series (Ch 11) What is a sequence? A series? Determining Series Convergence Divergence Test Integral Test Comparison Tests Alternating Series Test Ratio Test Nth-Root Test Power Series Intervals and Radii of Convergence New Functions from Old Taylor Series and Maclaurin Series Thursday, Nov 10 MAT 146

Distance Walked (each step) Xeno’s Paradox Step # Distance Walked (each step) Total Distance Walked 1 10 10= 10/(1/2) 2 5 10 + 5 = 15= 15/1 3 5/2 10 + 5 + 5/2 = 17 1/2 = 35/2 4 5/4 10 + 5 + 5/2 + 5/4 = 18 3/4 = 75/4 5/8 10 + 5 5/2+ 5/4+ 5/8 = 19 3/8= 155/8 … n Thursday, Nov 10 MAT 146

Sequence Characteristics Convergence/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? Thursday, Nov 10 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 the infinite number of terms in the sequence {an}. Thursday, Nov 10 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. Thursday, Nov 10 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? Thursday, Nov 10 MAT 146

Notable Series The harmonic series is the sum of all possible unit fractions. Thursday, Nov 10 MAT 146

Notable Series A telescoping sum can be compressed into just a few terms. Thursday, Nov 10 MAT 146

Fact or Fiction? Thursday, Nov 10 MAT 146

Why Study Sequences and Series in Calc II? Taylor Polynomials applet Infinite Process Yet Finite Outcome . . . How Can That Be? Transition to Proof Re-Expression! Thursday, Nov 10 MAT 146

Polynomial Approximators Our goal 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: Thursday, Nov 10 MAT 146

Thursday, Nov 10 MAT 146

Thursday, Nov 10 MAT 146

Thursday, Nov 10 MAT 146

Polynomial Approximators Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for Thursday, Nov 10 MAT 146

Create a 3rd-degree polynomial approximator for Thursday, Nov 10 MAT 146