1. MAT 1236 Calculus III Section 14.5 The Chain Rule http://myhome.spu.edu/lauw

2. HW.. WebAssign 14.5 Quiz: 14.5

3. Preview The Chain Rule for multivariable functions Look only at how to memorize the formulas Why the formulas actually work depends on 14.4 which we do not cover.

4. The Doppler Effect

5. 1 variable Vs 2 variables

6. The Chain Rule

7. The Chain Rule: Case 1

8. Example 1 Find

9. The Chain Rule: Case 2

10. Example 2 Find

11. Other Cases Similar

12. Example 3 Find in terms of partial derivatives in x and y (i.e. ) Note that the function f is not given explicitly.

13. Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation What is the relationship between

14. Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation What is the relationship between

15. Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation

16. Example 4 (Implicit Differentiation) Suppose y is a function in x and is given implicitly by the equation Show that

17. Where to use? (If time permitted) Taken from MAT 3724 Applied Analysis (Mathematical Physics)

18. Wave Equation: Set Up Assumptions: 1. 2. 3.

19. Thought Experiment

20. What would you expect to happen?

21. Thought Experiment

23. Modeling Wave Propagation in a String

25. Solution: d’Alembert’s Formula