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## AP Exam Review Competition

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**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!**Answer:***green terms 0 1.Find the limit. Recall key step: divide all terms by the highest power x3**Answer:**2.Find the derivative.**Answer:**3. Evaluate:**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**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).**6. Evaluate:**Answer: Answer: 7. Evaluate f ’(x): 8. Name the theorem used in problem 7 above. Answer: Fundamental Thrm of Calculus**(-1,3)**(5,3) Answer: 9 (Two triangles with b = 3 and h = 3.) (2,0) 9. Evaluate:**Answer:**10. Differentiate with respect to t (time): PV = c where c is a constant**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?**13. Evaluate. (You must have both correct!)**Answer: tan-1x + C and sin-1x + C**Answer:**14. Find the derivative. Recall key step: apply the quotient rule**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)**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)**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**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**Answer:**(by FTC 2) 19. Find f’(x) if:**Answer:**20. Evaluate:**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.**(3, 1)**Answer: jump discontinuity (3, -1) 22.Name the type of discontinuity at x = 3 for**Answer:**(by FTC 2) 23. Find f’(x) if:**Answer:**24. Evaluate:**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:**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:**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.**Answer:**28. Evaluate:**Answer:**29. Evaluate:**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**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**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 = ?**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**Answer:**34. Evaluate both:**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)**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)**½**–1 37. Given the graph of f(x), evaluate Answer: – ½ Sum of 2 Δs ½ + -1 f(x)**Answer: 0**(it is an odd function) 38. Evaluate:**Answer:**39. Give the third part of the definition of continuity: “f is continuous at c if: i. ii. iii. ???**Answer:**40. Find the derivative (and factor the GCF).**Answer:**41. Evaluate:**Answer:**42. Find the equation of the tangent to y = x3 – 1 at x =1.**Answer:**43. Evaluate:**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?**Answer:**When degrees are equal the horiz asymptote is at ratio of leading coefficients. 45. Evaluate:**Answer: - ∞**Next to vertical asymp, think of the signs of num. over denom. 46. Evaluate:**47. Find the slope of the normal to**y = x3 – 1 at x =1. Answer:**Answer:**48. Evaluate:**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**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.**Answer: 12**Shortcut: this is def of derivative for f ’(2) where f(x) = x3, so use f ’(x)= 3x2 Or Alg: 51. Evaluate:**Answer:**52. Evaluate: