Lesson 3-1: Derivatives

1. Lesson 3-1: Derivatives AP Calculus Mrs. Mongold

2. We write: There are many ways to write the derivative of is called the derivative of at . “The derivative of f with respect to x is …”

3. “the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” or “dee why dee ecks” “the derivative of f with respect to x” or “dee eff dee ecks” “the derivative of f of x” “dee dee ecks uv eff uv ecks” or

4. Note: dx does not mean d times x ! dy does not mean d times y !

5. does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)

6. does not mean times ! Note: (except when it is convenient to treat it that way.)

7. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

9. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

10. Example • Differentiate f(x)=x3

11. Example • Differentiate f(x)=x3 To differentiate we take the limit

12. Example • Differentiate f(x)=x3 To differentiate we take the limit

13. Example • Differentiate f(x)=x3 To differentiate we take the limit

14. Example • Differentiate f(x)=x3 To differentiate we take the limit

15. Example • Differentiate f(x)=x3 To differentiate we take the limit

16. Example • Differentiate f(x)=x3 To differentiate we take the limit

17. Example • Differentiate f(x)=x3 To differentiate we take the limit

18. Example • Differentiate f(x)=

19. Example • Differentiate f(x)= So look at the limit

20. Example • Differentiate f(x)= So look at the limit

21. Example • Differentiate f(x)= So look at the limit

22. Example • Differentiate f(x)= So look at the limit

23. Example • Differentiate f(x)= So look at the limit

24. Example • Differentiate f(x)= So look at the limit

25. Example • Differentiate f(x)= So look at the limit

26. Alternate Definition • Derivative at a point • The derivative of a function f at the point x=a is the limit Provided the limit exists

27. Example • Use Alt. Def. to differentiate f(x)= at x=a

28. Example • Use Alt. Def. to differentiate f(x)= at x=a • Alt. Def. Limit is

29. Example • Use Alt. Def. to differentiate f(x)= at x=a • Alt. Def. Limit is

30. Example • Use Alt. Def. to differentiate f(x)= at x=a • Alt. Def. Limit is

31. Example • Use Alt. Def. to differentiate f(x)= at x=a • Alt. Def. Limit is

32. Example • Use Alt. Def. to differentiate f(x)= at x=a • Alt. Def. Limit is Domain (0,

33. Example • Use Alt. Def. to differentiate f(x)= at a=2

34. Example • Use Alt. Def. to differentiate f(x)= at a=2 • Previous example gave us

35. Example • Use Alt. Def. to differentiate f(x)= at a=2 • Previous example gave us • So when a = 2 then

36. Lots to Remember with derivatives • The derivative is the slope at a point • When graphing a derivative the x values stay the same, but the y-values for the graph of f’ are the slopes from the points on f • So positive slope means f’ graph is above x axis • So negative slope means f’ graph is below x axis • 0 slope means f’ graph crosses the x axis

37. Example • Graph the derivative of f

38. Example • Graph the derivative of f

39. Example • Graph the derivative of f

40. Example • Graph the derivative of f

41. Example • If f(x) = x3-x, find a formula for f’(x) and illustrate by comparing f and f’ graphs

42. Example • Graph f from f’ • Sketch a graph of a function f that has the following properties • f(0)=0, f( • The graph of f’, the derivative of f, is below on left • F is continuous for all x

43. Example • Graph f from f’ • Sketch a graph of a function f that has the following properties • f(0)=0 • The graph of f’, the derivative of f, is below on left • F is continuous for all x

44. Example • Sketch the graph of a continuous function f , with f(0) = -1 and

45. Example • Sketch the graph of a continuous function f , with f(0) = -1 and -1

46. Homework • Page 101/ 1-12 in Blue and Red Calc Book