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Third Annual NCMATYC Math Competition November 16, 2013 Calculus Test Please do NOT open this booklet until given the signal to begin. You have 90 minutes to complete this 40-question multiple choice calculus test. Approximately two-thirds of the questions pertain to topics covered in a typical Calculus I course; the rest pertain to topics covered in a typical Calculus II course. Answer the questions on the electronic grading form by giving the best answer to each question. The scoring will be done by giving one point for each question answered correctly and zero points for each question answered incorrectly or left blank. Note that the scoring does not include a “guessing penalty” and so it is to your advantage to answer all the questions, even if you have to guess. If there is a tie, question number 19 will be used as a tie-breaker. This test was designed to be a CHALLENGE. There will surely be questions that you will not be able to answer. We hope you will enjoy working on and take pride in those questions which you are able to answer. You may write in the test booklet. You may keep your test booklet and any of your scrap papers. Only the electronic grading form will be collected and graded. Good luck! NCMATYC Calculus Tournament Fall 2013 1. Suppose f and g are differentiable functions for all real numbers and h x f g x . Furthermore, it is known that g 1 4; g 1 5; g 4 2; f 1 4; f (1) 9; f (4) 1. Determine h 1 . A. 5 B. 16 2. What is lim t 0 A. 3 4 C. 45 D. 16 E. 18 C. D. 0 E. does not exist 27 3t 2 27 ? 4t B. 3 4 3. What is the 20th derivative of the function f x cos x sin x ? A. f 20 x 220 cos x 219 2 B. f 20 x 220 cos x sin x D. f 20 x 219 220 cos x E. f 2 20 C. f 20 x 220 cos x sin x x 220 cos2 x sin 2 x 4. Let f and g be differentiable functions for all real numbers, what is lim f g x h f g x h 0 A. f g x B. f g x C. f g x D. f g x h ? E. none of these 5. Which of the following types of functions must have a derivative equal to 0 at some value in its domain? A. Cubic B. Quartic C. Exponential 1 D. Tangent E. none of these NCMATYC Calculus Tournament Fall 2013 6. Let f and g be differentiable functions such that g x x 3 f x 2 1 , f 5 3 , and f 5 2 . What is the value of g 2 ? A. −12 B. −18 7. What is lim x 0 A. B. undefined 8. Determine A. E. 0 C. 1 2 D. 0 E. none of these d2y 2 2 D. 2 dx 9x E. d2y for y 3 x 2 . dx 2 d 2 y y 4x A. dx 2 x2 D. 12 x ? x x 1 2 9. Compute C. 20 d 2 y y 4x B. dx 2 3y2 d 2 y y 2x C. dx 2 9 y2 d2y 2 2 2 dx 9y 1 ln x dx x2 1 ln x 1 C x D. B. 1 ln x C x 1 ln x 1 C x3 C. E. 1 ln x C x2 10. Which of the following is an inflection point on the graph of y A. There is no inflection point 10 B. 8, 49 2 C. 4, 25 2 1 3 2 ln x C x3 x2 ? ( x 1) 2 2 D. 8, 27 E. none of these NCMATYC Calculus Tournament Fall 2013 11. What is the equation of the tangent line to the graph of y A. y 7 x 2 B. y x 2 1 12. Evaluate: lim 1 x x ( x 2)3 ( x 1)2 at x 0 ? ( x 2)2 C. y 2 x 2 D. y 7 x 2 C. e2 D. 0 E. none of these 2x A. e B. 2e E. 13. For how many integer values of c does lim x x equal a real number? x c A. 1 B. 0 C. 2 D. infinitely many E. impossible to determine 14. For how many values of the domain does the function f x xe x cos x have a horizontal tangent line? A. 0 B. 1 C. 2 D. 4 E. infinitely many 15. Which of the following is the derivative of the function F x f x g x h x ? i. F x f x g x f x g x h x f x g x h x ii. F x g x h x g x h x f x f x g x h x iii. F x f x h x f x h x g x f x g x h x A. i B. ii C. iii D. none of these 3 E. all of these NCMATYC Calculus Tournament Fall 2013 16. Evaluate 2 0 x 2 1dx A. 2 B. 2 3 C. 17. Evaluate the first derivative of y x dy sin x A. cos x ln x dx x D. 18. Define h x D. 3 8 3 E. none of these . 1 B. dy sin x cos x ln x xsin x dx x dy xsin x 1 sin x dx dy sin x cos x ln x dx x 2 3 x2 sin x 4 3 E. dy x cos x sin x dx 1 r dr . Evaluate h x . 2 A. h x 6 x 3 1 9 x 4 B. h x 6 x 3 1 9 x 4 E. h x 2 x 3 1 9 x 4 D. h x 3 1 9 x 4 19. Compute the local extrema of y A. No local extrema for x 3 C. h x 3 1 9 x4 5ln x ln ln x 2 for x 3 . 10 B. e 2 , ln 2 C. e 5 , ln 5 1 D. e e ,5e E. none of these 20. Suppose the radius of a cylinder is measured at 2 cm with a possible error of ±0.01 cm and the height is measured at 5 cm with a possible error of ± 0.1 cm. Estimate the error in the calculated volume using differentials to the nearest hundredth of a cubic centimeter. A. 2.2π cm3 B. 2.4π cm3 C. 0.6π cm3 4 D. 0.8π cm3 E. 1.0π cm3 NCMATYC Calculus Tournament Fall 2013 21. Let the quadratic function f x ax2 bx c be non-negative on the interval 1,1 . Which of the following is A. f x dx ? 1 1 2 f 1 2 f 0 f 1 3 D. B. 1 f 1 4 f 0 f 1 2 1 f 1 4 f 0 f 1 3 C. 1 f 1 2 f 0 f 1 4 E. None of these 22. Which of the following is represents the volume of the solid formed by revolving the region bounded by the graphs of y ln x , x-axis, and the line x 3 about the line x 3 . A. 9 ln x dx 3 1 2 B. ln 3 0 9 e dy D. ln x dx 3 C. 2y ln 3 0 3 e dy 2 y 2 E. None of these 1 23. The intersection of y x 2 14 and y x creates two tangent lines, one to each curve. Which of the following expressions gives the angle created by the intersection of these two tangent lines? 1 A. tan 1 2 tan 1 2 1 B. tan 1 8 tan 1 4 1 D. tan 1 8 tan 1 8 1 C. tan 1 4 tan 1 4 1 E. tan 1 4 tan 1 2 24. L be the tangent line to x 2 3x 2 y y 3 5 at the point 1,1 . What is the sum of the intercepts of L? A. 5 6 B. 49 12 C. 7 12 5 D. 5 6 E. 11 30 NCMATYC Calculus Tournament Fall 2013 25. What is the arc length of y e0.5 x e0.5 x from x 0 to x 2ln 3 ? A. 2 3 B. 4 3 C. 8 3 D. 16 3 E. 32 3 26. Determine the minimal distance from the point 5,0 to the graph y x 2 1 . A. 15 B. 4 5 C. D. 34 E. 2 5 26 27. A circle centered at the origin passes through the point 4, 2 . Determine the area in Quadrant IV between the tangent line to the circle at 4, 2 and the circle. A. 16 5 C. 32 10 B. 32 5 D. 50 10 E. 25 5 28. Let R be the region in Quadrant I bounded by y 1 x3 , the line y 0 , and the line x 0 . Calculate the volume of the solid of revolution generated by revolving R about the line x 1 . A. 9 10 29. What is the value of A. 1 B. 9 14 C. 4 x2 0 x2 4 x B. 2 2 5 14 D. 3 5 E. 5 3 dx ? C. 4 D. 6 6 E. 8 NCMATYC Calculus Tournament Fall 2013 30. Determine A. d2y if 𝑦 4 − 2𝑦 = 𝑥 dx 2 12 y 2 4 y3 2 2 B. 12 y 2 4 y3 2 C. 3 12 y 2 4 y3 2 2 D. 12 y 2 4 y3 2 3 E. None of these 31. A particular radio station plays eight minutes of music followed by two minutes of commercials. If a listener tunes in randomly, what is the average number of minutes he will listen before he hears a commercial? A. 3.2 min B. 4 min C. 2 min D. 2.8 min E. 3.6 min 32. Determine the area of the largest rectangle that can be inscribed in a semicircle of radius 4 cm. A. 8 cm2 B. 16 cm2 C. 12 cm2 D. 10 cm2 E. 32 cm2 33. Determine the equation of the normal line to the curve y x 4 1 ln x 1 at x 0 . 3 B. 2 y x A. y x 34. Evaluate A. 16 e 0 D. y x E. y 2 x ecost sin 2t dt . B. 4e 35. What is the length of the curve y A. 2 2 C. y 2 x B. 1 C. 2 e 1 D. 4 e 1 D. 2 E. 4 e x x sin x ? 2 1 C. 2 7 E. none of these NCMATYC Calculus Tournament Fall 2013 36. Let 2 dv 2tv 3t 2 et and v 0 8 . What is v t ? dt A. v t t 2et 8et 2 2 B. v t tet 8et 2 2 C. v t t 3et 8et 2 D. v t 8et 2 2 E. none of these 1 2n 37. Evaluate n . n 1 9 A. It is divergent. B. 1 5 C. 15 42 D. 3 8 E. 38. Which of the following is the general solution for the differential equation x 3e x B. y x 4 x 3e x A. y dx C x D. y x 4 dx C 3xe dx C 23 56 dy 4 y 3x 4e x ? dx C. y 3 x 4 e x dx C E. y 3xe x dx C x 39. A rope is 40 feet long and weighs 0.5 lb/ft. If it hangs over the edge of a building 200 feet tall, how much work is done by pulling half of the rope to the top of the building? A. 240 ft-lb B. 300 ft-lb C. 260 ft-lb D. 380 ft-lb E. none of these 40. Find the centroid of the region bounded by the curve 4x2 – 24x + y2 + 10y + 45 = 0 A. (0, 0) B. (1, -3) C. (2, -4) 8 D. (3, -5) E. (4, -6)