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

2. HW.. WebAssign 14.5 (14 problems, 184min.) 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

13. Remark Do this once on your own before you attempt your HW.

14. Example 4 (Implicit Differentiation)

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

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

20. Thought Experiment

21. What would you expect to happen?

22. Thought Experiment

24. Modeling Wave Propagation in a String

26. Solution: d’Alembert’s Formula