Calculus Date: 12/17/13 Obj : SWBAT apply first derivative test

1. Calculus Date:12/17/13 Obj: SWBAT apply first derivative test http://youtu.be/PBKnttVMbV4 first derivative test inc.dec. Today – Cover the first derivative test In class: Start WS3-3A; complete for homework Tomorrow Cover the second derivative test (easier than 1st) Friday: With Christian complete any remaining worksheets, make sure You understand material from the test.,etc. • Announcements: • Break Packet online on Friday • Merry Christmas if I don’t see you "Do not judge me by my successes, judge me by how many times I fell down and got back up again.“ Nelson Mandela

2. Increasing/Decreasing/Constant

3. Increasing/Decreasing/Constant

4. Increasing/Decreasing/Constant

5. A similar Observation Applies at a Local Max. Generic Example The First Derivative Test

6. The First Derivative Test Determine the sign of the derivative of f to the left and right of the critical point. left right conclusion f (c) is a relative maximum f (c) is a relative minimum No change No relative extremum

7. Relative max. f (0) = 1 Relative min. f (4) = -31 The First Derivative Test Find all the relative extrema of + 0 - 0 + 0 4 Evaluate the derivative at points on either side of extrema to determine the sign.

8. The First Derivative Test Sketch of function based on estimates from the first derivative test.

9. The First Derivative Test Here is the actual function.

10. Another Example Find all the relative extrema of Stationary points: Singular points:

11. Stationary points: Singular points: Evaluate the derivative at points on either side of extrema to determine the sign. Use Yvars for faster calculations ND – derivative not defined Relative max. Relative min. + ND + 0 - ND - 0 + ND + -1 0 1

12. Local min. Local max. Graph of + ND + 0 - ND - 0 + ND + -1 0 1

13. There are roots at and . Possibleextremaat . Set Example: Graph We can use a chart to organize our thoughts. First derivative test: y y=4 y=0 positive negative positive

14. There are roots at and . Possible extreme at . Set maximum at minimum at Example: Graph First derivative test: y y=4 y=0

15. There is a local maximum at (0,4) because for all x in and for all x in (0,2) . There is a local minimum at (2,0) because for all x in (0,2) and for all x in . Example: Graph NOTE: On the AP Exam, it is not sufficient to simply draw the chart and write the answer. You must give a written explanation! First derivative test:

16. There is a local maximum at (0,4) because for all x in and for all x in (0,2) . There is a local minimum at (2,0) because for all x in (0,2) and for all x in . Example: Graph First derivative test:

17. Graph • Take the derivative f’(x) • Find the critical points f’(x) = 0; f’(x) = DNE • Make sign chart • Label critical points – put 0 or DNE on graph • Evaluate the derivative at points on either side of extrema to determine the sign. Use Yvars for faster calculations. Mark the chart as + or – in these areas • Evaluate f(x) at the critical points to determine actual values of extrema. • Add arrows to show increasing or decreasing regions in f(x) • Write out all extrema and use the value of the derivative and the appropriate interval to justify your answer.

18. Chapter 5Applications of the DerivativeSections 5.1, 5.2, 5.3, and 5.4

19. Applications of the Derivative • Maxima and Minima • Applications of Maxima and Minima • The Second Derivative - Analyzing Graphs

20. Absolute Extrema Let f be a functiondefined on a domain D Absolute Maximum Absolute Minimum

21. Absolute Extrema A function f has an absolute (global) maximum atx = c if f (x) f (c)for allx in the domain D of f. The number f (c) is called the absolute maximumvalue of f in D Absolute Maximum

22. Absolute Extrema A function f has an absolute (global) minimum atx = c if f (c) f (x)for allx in the domain D of f. The number f (c) is called the absolute minimumvalue of f in D Absolute Minimum

23. Generic Example

24. Generic Example

25. Generic Example

26. Relative Extrema A function f has a relative (local) maximum at xc if there exists an open interval (r, s) containing c suchthat f (x) f (c) for all r  x  s. Relative Maxima

27. Relative Extrema A function f has a relative (local) minimum at xc if there exists an open interval (r, s) containing c suchthat f (c) f (x) for all r  x  s. Relative Minima

28. Generic Example The corresponding values of x are called Critical Points of f

29. Critical Points of f A critical number of a function f is a number cin the domain off such that (stationary point) (singular point)

30. Candidates for Relative Extrema • Stationary points: any x such that xis in the domain of f and f'(x)  0. • Singular points: any x such that xis in the domain of f and f'(x)  undefined • Remark:notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

31. Fermat’s Theorem If a function f has a local maximum or minimum at c, then c is a critical number of f Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum

32. Generic Example Two critical points of f that do not correspond to local extrema

33. Example Find all the critical numbers of Stationary points: Singular points:

34. Local min. Local max. Graph of

35. Extreme Value Theorem If a function f is continuous on a closed interval [a,b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint. a b a b a b Attains max. and min. Attains min. but no max. No min. and no max. Open Interval Not continuous

36. Absolute Max. Absolute Min. Absolute Max. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Evaluate

37. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max. Absolute Min.

38. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Evaluate Absolute Min.

39. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Absolute Min.

40. Start Herea. Reviewing Rolle’s and MVTb. Remember nDeriv and Yvarsc. Increasing, Decreasing,Constantd. First Derivative Test

41. Finding absolute extrema on [a,b] 0. Verify function is continuous on the interval. Determine the function’s domain. • Find all critical numbers for f (x) in (a,b). • Evaluate f (x) for all critical numbers in (a,b). • Evaluate f (x) for the endpoints a and b of the interval [a,b]. • The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a,b].

42. c Rolle’s Theorem • Given f(x) on closed interval [a, b] • Differentiable on open interval (a, b) • If f(a) = f(b) … then • There exists at least one numbera < c < b such that f ’(c) = 0 f(a) = f(b) b a

43. The Mean Value Theorem (MVT) aka the ‘crooked’ Rolle’s Theorem f(b) a c b f(a) • We can “tilt” the picture of Rolle’s Theorem • Stipulating that f(a) ≠ f(b) If f is continuous on [a, b] and differentiable on (a, b) There is at least one number c on (a, b) at which Conclusion: Slope of Secant Line Equals Slope of Tangent Line How is Rolle’s Connected to MVT?

44. The Mean Value Theorem (MVT) aka the ‘crooked’ Rolle’s Theorem f(b) a c b f(a) • We can “tilt” the picture of Rolle’s Theorem • Stipulating that f(a) ≠ f(b) If f is continuous on [a, b] and differentiable on (a, b) There is at least one number c on (a, b) at which Conclusion: The average rate of change equals the instantaneous rate of change evaluated at a point

45. Finding c • Given a function f(x) = 2x3 – x2 • Find all points on the interval [0, 2] where • Rolle’s? • Strategy • Find slope of line from f(0) to f(2) • Find f ‘(x) • Set f ‘(x) equal to slope … solve for x

46. If , how many numbers on [-2, 3] satisfy the conclusion of the Mean Value Theorem. A. 0 B. 1 C. 2 D. 3 E. 4 CALCULATOR REQUIRED f(3) = 39 f(-2) = 64 For how many value(s) of c is f ‘ (c ) = -5? X X X

47. Given the graph of f(x) below, use the graph of f to estimate the numbers on [0, 3.5] which satisfy the conclusion of the Mean Value Theorem.

48. Relative Extrema Example:Find all the relative extrema of Stationary points: Singular points: None

49. First Derivative Test • What if they are positive on both sides of the point in question? • This is called aninflection point 

50. Domain Not a Closed Interval Example: Find the absolute extrema of Notice that the interval is not closed. Look graphically: Absolute Max. (3, 1)