Integration Applications l Area Between Curves 6 1
Integration Applications l Area Between Curves (6. 1) l Average Value of a Function (6. 5) l Volumes of Solids (6. 2, 6. 3) l Created by Rotations l Created using Cross Sections l Arc Length of a Curve (8. 1) l Probability (8. 5) Methods of Integration l U-substitution (5. 5) l Integration by Parts (7. 1) l Trig Integrals (7. 2) l Trig Substitution (7. 3) l Partial-Fraction Decomposition (7. 4) l Putting it All Together: Strategies! (7. 5) l Improper Integrals (7. 8) Wednesday, May 2, 2018 Differential Equations l What is a differential equation? (9. 1) l Solving Differential Equations l Visual: Slope Fields (9. 2) l Numerical: Euler’s Method (9. 2) l Analytical: Separation of Variables (9. 3) l Applications of Differential Equations Infinite Sequences & Series (Ch 11) l What is a sequence? A series? (11. 1, 11. 2) l Determining Series Convergence l Divergence Test (11. 2) l Integral Test (11. 3) l Comparison Tests (11. 4) l Alternating Series Test (11. 5) l Ratio Test (11. 6) l Nth-Root Test (11. 6) l Power Series l l l Interval & Radius of Convergence New Functions from Old Taylor Series and Maclaurin Series
l If we let cn = 1 for all n, we get a familiar series: l This geometric series has common ratio x and we know the series converges for |x| < 1. l We also know the sum of this series: Wednesday, May 2, 2018
Wednesday, May 2, 2018
Wednesday, May 2, 2018
Wednesday, May 2, 2018
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: Wednesday, May 2, 2018
Wednesday, May 2, 2018
Goal: Generate polynomial functions to approximate other functions near particular values of x. Create a third-degree polynomial approximator for Wednesday, May 2, 2018
Wednesday, May 2, 2018
Wednesday, May 2, 2018
Create a 3 rd-degree polynomial approximator for Wednesday, May 2, 2018
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! Wednesday, May 2, 2018
Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3 Taylor Series Demo #4 Wednesday, May 2, 2018
Look at characteristics of the function in question and connect those to the cn. Example: f(x) = ex, centered around a = 0. Wednesday, May 2, 2018
Example: f(x) = ex, centered around a = 0. And…how far from a = 0 can we stray and still find this re-expression useful? Wednesday, May 2, 2018
General Form: Coefficients cn Wednesday, May 2, 2018
Examples: Determining the cn f(x) = cos(x), centered around a = 0. Wednesday, May 2, 2018
Examples: Determining the cn f(x) = sin(x), centered around a = 0. Wednesday, May 2, 2018
Examples: Determining the cn f(x) = ln(1 -x), centered around a = 0. Wednesday, May 2, 2018
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