Calculus II (MAT 146) Dr. Day Thursday, Dec 8, 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) Semester Exam Wednesday, Dec 14 (STV 229) 10 am – 12:40 pm Review Session Monday, Dec 12, 5:45-6:45 pm (STV 346) 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 and p-Series 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, Dec 8, 2016 MAT 146

Absolute Convergence and Conditional Convergence Thursday, Dec 8, 2016 MAT 146

Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does each series converge? Determine the Radius of Convergence and the Interval of Convergence for each power series. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Use the Ratio Test to determine values of x that result in a convergent series. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Determine its Radius of Convergence and its Interval of Convergence. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Determine its Radius of Convergence and its Interval of Convergence. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Use the Ratio Test to determine values of x that result in a convergent series. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Determine its Radius of Convergence and its Interval of Convergence. Thursday, Dec 8, 2016 MAT 146

Power Series Convergence For what values of x does this series converge? Determine its Radius of Convergence and its Interval of Convergence. Thursday, Dec 8, 2016 MAT 146

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: Thursday, Dec 8, 2016 MAT 146

Geometric Power Series Thursday, Dec 8, 2016 MAT 146

Geometric Power Series Thursday, Dec 8, 2016 MAT 146

Geometric Power Series Thursday, Dec 8, 2016 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, Dec 8, 2016 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, Dec 8, 2016 MAT 146

Thursday, Dec 8, 2016 MAT 146

Thursday, Dec 8, 2016 MAT 146

Thursday, Dec 8, 2016 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, Dec 8, 2016 MAT 146

Create a 3rd-degree polynomial approximator for Thursday, Dec 8, 2016 MAT 146