AP Exam Review Competition

1. AP Exam Review Competition • First to finish calls “15 seconds”. • Answers must be written down & then revealed. • Add 1 pt to your score for each correct response. • Come back later for review & practice!

2. Answer: *green terms 0 1.Find the limit. Recall key step: divide all terms by the highest power x3

3. Answer: 2.Find the derivative.

4. Answer: 3. Evaluate:

5. 4. Fill in the blanks: Since polynomial functions are continuous over the reals and for f(x) = x3 -1, we know f(0) = -1 and f(2) = 7, there exists a value c in the interval ________such that f(c)=5 by the ___________________ Theorem. Answer: (0, 2) Intermediate Value

6. 5. Given f(x) and g(x) are diff. over R and g(x) = f -1(x). If f(5)=7, f ’(5)=2, f(9)=5 and f ’(9)=6, find g’(5). Answer: 1/6 Slopes on inverses are reciprocals at corresponding pts. Since (9,5) is on f, then (5, 9) is on g . . . so we simply take the reciprocal of f’(9) to get g’(5).

7. 6. Evaluate: Answer: Answer: 7. Evaluate f ’(x): 8. Name the theorem used in problem 7 above. Answer: Fundamental Thrm of Calculus

8. (-1,3) (5,3) Answer: 9 (Two triangles with b = 3 and h = 3.) (2,0) 9. Evaluate:

9. Answer: 10. Differentiate with respect to t (time): PV = c where c is a constant

10. Answer: Answer: 11. For s(t) = t2 + 1, what is the average velocity over the time interval (0,4) seconds if distance is given in ft? 12. For s(t) = t2 + 1 above, what is the instantaneous velocity at t = 4?

11. 13. Evaluate. (You must have both correct!) Answer: tan-1x + C and sin-1x + C

12. Answer: 14. Find the derivative. Recall key step: apply the quotient rule

13. 3 f ' (x) 2 (a,b) 1 0 3 1 2 2 4 -1 -2 (c,d) -3 15. Given the graph of f ‘(x) shown, give the x-coordinate(s) where f(x) has local minima. Answer: 0 and 3 (where slopes change from neg to pos)

14. 3 f ' (x) 2 (a,b) 1 0 3 1 2 2 4 -1 -2 (c,d) -3 16. Given the graph of f ‘(x) shown, give the x-coordinates where f(x) has points of inflection. Answer: a and c (where f ‘changes from incr  decr, the concavity will change)

15. 17. If f(x) = g ( h(x) ), then f ’(x) = __?__ Answer: f ’(x) = g’(h(x)) ● h’(x) *Derivative of a composite function requires the chain rule

16. 18.By the 2nd Derivative Test, if f ”(x) is continuous, f ’(2) =0, and f ”(2) > 0, then (2, f(2) ) is a ____ ____. Answer: local minimum. *horiz. tangent in a concave up interval  local min

17. Answer: (by FTC 2) 19. Find f’(x) if:

18. Answer: 20. Evaluate:

19. 21. If f(x) = sin2(3x), find f’(x). Answer: f ’(x) = 2sin(3x) ● cos(3x) ● 3 *Power rule and two chain rules on the inside functions.

20. (3, 1) Answer: jump discontinuity (3, -1) 22.Name the type of discontinuity at x = 3 for

21. Answer: (by FTC 2) 23. Find f’(x) if:

22. Answer: 24. Evaluate:

23. Answer: Recall this is the height of a rectangle that has the same area as the area under the curve. 25. Find the average value of f(x) over the interval (2,7) given:

24. Answer: -5, -1, 0, 1 *Crit # are where f ’(x) = 0  -1, 0, 1 or f ’(x) is undefined  -5 26. Find the critical #s of f(x) if:

25. 27. If oil leaks from a tank at a rate of r(t) gallons per minute what does represent? Answer: the net number of gallons that leaked from the tank in the first five minutes.

26. Answer: 28. Evaluate:

27. Answer: 29. Evaluate:

28. 30. If f(x) is differentiable over the reals & f ”(x)=(x –1)(x –2), over which interval(s) is f(x) concave down? Answer: (1, 2) f ” < 0  concave down (-∞ ,1) (1, 2) (2, ∞) f”(x)>0 f ”(x) <0 f ”(x) >0

29. 31. If f is continuous at (c, f(c)), which of the following could be FALSE? A. B. C. D. Answer: C (e.g., a corner is continuous, but not differentiable) A, B & D are the very def of continuous

30. Answer: when t=3 and t=5 seconds When v(t) = 0 32. A particle moves along the x-axis so that its position at any time t  0 is given by The particle is at rest when t = ?

31. 33. P(t) = 520e570t is the model for the number of fruit flies at time t hours in a biology experiment. What do you know about the population at t = 0 hours? Answer: 520 fruit flies For P(t) = Cekt, C is the initial pop

32. Answer: 34. Evaluate both:

33. 3 f (x) 2 (a,b) 1 0 3 1 2 2 4 (c,d) -1 -2 (e,f) -3 35. Given the graph of f (x) shown, find the interval(s) where f ”(x) < 0. Answer: (-∞, c) (where f is concave down)

34. 3 f (x) 2 (a,b) 1 0 3 1 2 2 4 (c,d) -1 -2 (e,f) -3 36. Given the graph of f (x) shown give the interval(s) where f ’(x) < 0. Answer: (a, e) (where f is decreasing)

35. ½ –1 37. Given the graph of f(x), evaluate Answer: – ½ Sum of 2 Δs ½ + -1 f(x)

36. Answer: 0 (it is an odd function) 38. Evaluate:

37. Answer: 39. Give the third part of the definition of continuity: “f is continuous at c if: i. ii. iii. ???

38. Answer: 40. Find the derivative (and factor the GCF).

39. Answer: 41. Evaluate:

40. Answer: 42. Find the equation of the tangent to y = x3 – 1 at x =1.

41. Answer: 43. Evaluate:

42. Answer: Since f is differentiable over (1,3) and the slope of the secant between endpoints is , there exists a value c in (1,3) such that f’(c) = 2. 44. If f is differentiable over (1, 3), f(1)=4, and f(3)= 8, what can you conclude by the Mean Value Theorem?

43. Answer: When degrees are equal the horiz asymptote is at ratio of leading coefficients. 45. Evaluate:

44. Answer: - ∞ Next to vertical asymp, think of the signs of num. over denom. 46. Evaluate:

45. 47. Find the slope of the normal to y = x3 – 1 at x =1. Answer:

46. Answer: 48. Evaluate:

47. 49. If f is differentiable over (0,4) and f(1)=7 and f(3)=5, then we know there exists a c in ___?___ such that f(c)=6 by the _________?_________. Answers: (1,3) Intermediate Value Theorem

48. 50. If f is differentiable over (0,4) and we know f(2) = 7 and f’(2)=3, what is the best approximation we can give for f(2.1)? Answers: 7.3 by Linear Approx. Tangent line is: y – 7 = 3(x – 2) *Find (2.1, ?) on tangent as a close approximation since the tangent lies close to the f(x) curve.

49. Answer: 12 Shortcut: this is def of derivative for f ’(2) where f(x) = x3, so use f ’(x)= 3x2 Or Alg: 51. Evaluate:

50. Answer: 52. Evaluate: