1. Rules of differentiation REVIEW:

2. The Chain Rule

3. Taylor series

4. Approximating the derivative

5. Monday Sept 14th: Univariate Calculus 2 Integrals ODEs Exponential functions

6. Antiderivative (indefinite integral)

8. Area under a curve = definite integral

9. Integrating data: the trapezoidal rule Very similar!

10. Example: integrating a linear function

11. Another angle: the upper limit as an argument

14. Differential equations Algebraic equation: involves functions; solutions are numbers. Differential equation: involves derivatives; solutions are functions. INITIAL CONDITION

15. e.g. dead reckoning

16. Example

17. Classification of ODEs Linearity: Homogeneity: Order:

18. Superposition (linear, homogeneous equations) Can build a complex solution from the sum of two or more simpler solutions.

19. Superposition (linear, inhomogeneous equations)

21. Superposition (nonlinear equations)

22. ORDINARY differential equation (ODE): solutions are univariate functions PARTIAL differential equation (PDE): solutions are multivariate functions

23. 1 slope=1 Exponential functions: start with ODE Qualitative solution:

24. Exponential functions: start with ODE Analytical solution

25. Exponential functions: start with ODE Analytical solution

26. Rules for addition, multiplication, exponentiation

28. Differentiation, integration (chain rule)

29. Properties of the exponential function Sum rule: Power rule: Taylor series: Derivative Indefinite integral

30. Cooling

31. Sinking

33. Homework: Do exercises for section 2.6, 2.8 and 2.9. Omit 2.9, #1. This will include: Exercise with antiderivatives and classifying ODEs. Carbon dating (for Thursday field trip) Derive further well-known functions from f’’=-f