1. Final Review – Exam 4

2. Radius and Interval of Convergence (11.1 & 11.2) Given the power series, Refer to lecture notes or textbook of section 11.1 and 11.2 for more details.

3. Radius and Interval of Convergence (11.1 & 11.2) This occurs at the endpoints of the interval of convergence. So, the endpoints must be checked separately with some other tests to determine the interval of convergence of the series. Recall that the Ratio Test and the Root Test give no information when the limits in the tests is 1. Remark: if using the geometric series, then there is no need to check the endpoints.

4. Properties of Power Series (11.2) Remarks: a)We can obtain power series representations of functions by differentiating or integrating term by term known power series. b)The radius of convergence remains the same when a power series is differentiated or integrated. But the interval of convergence may change. Refer to lecture notes or textbook of section 11.2 for more details.

5. Example a)Differentiate the Taylor Series for f about 0. b)Identify the function represented by the differentiated series. c)Give the interval of convergence of the power series for the derivative.

6. Taylor Polynomials and Taylor Series (11.3)

7. Taylor Series and Maclaurin Series (11.3) Refer to lecture notes or textbook of section 11.3 for more details.

8. Working with Taylor Series (11.4) Differentiating Integrating Evaluating limits Refer to lecture notes or textbook of section 11.2 for details.

9. Work and Hydrostatic Force (6.7) Refer to lecture notes or textbook of section 6.7 for details. 0 y x 10 7

10. Hyperbolic Functions (7.7) Refer to lecture notes or textbook of section 7.7 for details. Review definitions of Hyperbolic Functions Derivatives of hyperbolic functions Integrals of hyperbolic functions

11. Parametric Equations (12.1) Derivative: Horizontal and vertical tangent lines: see lecture notes or textbook of section 12.1 for more details.

12. Polar Coordinates (12.2)  0 Refer to lecture notes or textbook of section 12.2 for more details.

13. Slopes of Tangent Lines (12.3)

14. Example

15. Area of Regions Bounded by Polar Curves Refer to lecture notes or textbook of section 12.3 for more details.

16. Example

19. Differential Equations (9.1 & 9.3) Refer to lecture notes or textbook of section 9.1 and 9.3 for details. Order of Differential Equations Linear or Nonlinear Solve initial value problems. Solve Differential Equations by separation of variables