MAT 1221 Survey of Calculus

1. MAT 1221Survey of Calculus Section B.1, B.2 Implicit Differentiation, Related Rates http://myhome.spu.edu/lauw

2. Expectations • Use equal signs • Simplify answers • Double check the algebra

3. HW • WebAssign HW B.1, B.2 • Additional HW listed at the end of the handout (need to finish, but no need to turn in) • Need to do your homework soon. Do not wait until tomorrow afternoon.

4. Exam 1 • You should have already started reviewing for Exam 1 • Proficiency: You need to know how to do your HW problem on your own • You need to understand how to solve problems • Memorizing the solutions of all the problems is not a good idea

5. Preview • Extended Power Rule Revisit • The needs for new differentiation technique –Implicit Differentiation • The needs to know the relation between two rates – Related Rates

6. Extended Power Rule

7. Extended Power Rule

8. Extended Power Rule

9. Extended Power Rule We now free the variable, which we need for the next formula.

10. Extended Power Rule • If is a function in , then • If y is a function in x, then

11. Extended Power Rule • If is a function in , then • If is a function in , then

12. Example 0

13. The Needs for Implicit Differentiation…

14. Example 1 • Find the slopes of the tangent line on the graph • i.e. find y x

15. Example 1: Method I • Make as the subject of the equation: y x

16. Example 1: Method I • Make as the subject of the equation: y x

17. Example 1: Method I • Make y as the subject of the equation: y x

18. Example 1: Method I Suppose the point is on the upper half circle y x

19. Example 1: Method I Suppose the point is on the lower half circle y x

20. Example 1: Method I • Two disadvantages of Method I: • ??? • ???

21. Example 1: Method II Implicit Differentiation: Differentiate both sides of the equation.

22. Example 2 Find the slope of the tangent line at

23. Notations

24. B.2. Related Rates

25. Related Rates • If and are related by an equation, their derivatives (rate of changes) and are also related.

26. Related Rates • If and are related by an equation, their derivatives (rate of changes) and are also related. • Note that the functions are time dependent • Extended Power Rule will be used frequently, e.g.

27. Example 3 • Consider a “growing” circle.

28. Example 3 • Both the radius and the area are increasing.

29. Example 3 • What is the relation between and ?

30. Example 3

31. Example 3 • A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

32. Step 1 Draw a diagram • A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

33. Step 2: Define the variables • A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

34. Step 3: Write down all the information in terms of the variables defined • A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

35. Step 4: Set up a relation between the variables

36. Step 5: Use differentiation to find the related rate Formal Answer When the radius is 5 feet, the area is changing at a rate of …

37. Steps for Word Problems 1. Draw a diagram 2. Define the variables 3. Write down all the information in terms of the variables defined 4. Set up a relation between the variables 5. Use differentiation to find the related rate. Formally answer the question.

38. Remark on Classwork #2 To save time, problem number 2 does not required all the steps.

39. Expectations • Use equal signs correctly. • Use and notations correctly. • Pay attention to the independent variables: Is it or ? • Pay attention to the units.