Calculus II (MAT 146) Dr. Day Monday April 30, 2018 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? (11.1,11.2) Determining Series Convergence Divergence Test (11.2) Integral Test (11.3) Comparison Tests (11.4) Alternating Series Test (11.5) Ratio Test (11.6) Nth-Root Test (11.6) Power Series Interval & Radius of Convergence New Functions from Old Taylor Series and Maclaurin Series Monday, April 30, 2018

Generalized Power Series This is called: a power series in (x – a), or a power series centered at a, or a power series about a. Monday, April 30, 2018

Power Series Convergence Use the Ratio Test! For what values of x does each series converge? Determine the Radius of Convergence and theInterval of Convergence for each power series. Monday, April 30, 2018

Geometric Power Series If we let cn = 1 for all n, we get a familiar series: This geometric series has common ratio x and we know the series converges for |x| < 1. We also know the sum of this series: Monday, April 30, 2018

Geometric Power Series Monday, April 30, 2018

Geometric Power Series Monday, April 30, 2018

Geometric Power Series Monday, April 30, 2018

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, April 30, 2018

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, April 30, 2018

Monday, April 30, 2018

Monday, April 30, 2018

Monday, April 30, 2018

Polynomial Approximators Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for Monday, April 30, 2018

Create a 3rd-degree polynomial approximator for Monday, April 30, 2018

Beyond Geometric Series Connections: Taylor Series How can we describe the cn so a power series can represent OTHER functions? ANY functions? Now we go way back to the ideas that motivated this chapter’s investigations and connections: Polynomial Approximators! Monday, April 30, 2018

Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3 Monday, April 30, 2018

Taylor Series Example: f(x) = ex, centered around a = 0. Look at characteristics of the function in question and connect those to the cn. Example: f(x) = ex, centered around a = 0. Monday, April 30, 2018

Taylor Series Example: f(x) = ex, centered around a = 0. And…how far from a = 0 can we stray and still find this re-expression useful? Monday, April 30, 2018

General Form: Coefficients cn Monday, April 30, 2018

Examples: Determining the cn f(x) = cos(x), centered around a = 0. Monday, April 30, 2018

Examples: Determining the cn f(x) = sin(x), centered around a = 0. Monday, April 30, 2018

Examples: Determining the cn f(x) = ln(1-x), centered around a = 0. Monday, April 30, 2018