Calculus II (MAT 146) Dr. Day Monday November 6, 2017 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 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) Monday, November 6, 2017

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? Monday, November 6, 2017 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}. It is a unique finite value, if it exists. 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. Monday, November 6, 2017 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? Monday, November 6, 2017 MAT 146

Notable Series The harmonic series is the sum of all possible unit fractions. Monday, November 6, 2017 MAT 146

Notable Series A telescoping sum can be compressed into just a few terms. Monday, November 6, 2017 MAT 146

Fact or Fiction? Monday, November 6, 2017 MAT 146

Our first Series Convergence Test Our first Series Convergence Test . . . The nth-Term Test also called The Divergence Test Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 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! Monday, November 6, 2017 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: Monday, November 6, 2017 MAT 146

Monday, November 6, 2017 MAT 146

Monday, November 6, 2017 MAT 146

Monday, November 6, 2017 MAT 146

Polynomial Approximators Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for Monday, November 6, 2017 MAT 146

Create a 3rd-degree polynomial approximator for Monday, November 6, 2017 MAT 146

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Monday, April 10, 2017 MAT 146

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Monday, April 10, 2017 MAT 146

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Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146

Converge or Diverge? Monday, April 10, 2017 MAT 146

Converge or Diverge? (9) Monday, April 10, 2017 MAT 146

Converge or Diverge? (12) Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146

Absolute Convergence and Conditional Convergence Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146

Monday, April 10, 2017 MAT 146