Third Annual NCMATYC Math Competition November 16, Calculus Test
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1 Third Annual NCMATYC Math Competition November 6, 0 Calculus Test Please do NOT open this booklet until given the signal to begin. You have 90 minutes to complete this 0-question multiple choice calculus test. Approimately 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 9 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!
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3 NCMATYC Calculus Tournament Fall 0. Suppose f and g are differentiable functions for all real numbers and is known that g ; g 5; g ; f ; f ( ) 9; f (). Determine h f g. Furthermore, it 5 B. 6 C. 5 D. 6 E. 8 h.. What is lim t0 7 t 7 t? B. C. D. 0 E. does not eist. What is the 0 th derivative of the function f cos sin? f 0 0 cos 9 B. 0 0 cos sin f C. 0 0 cos sin f D cos f E. 0 0 cos sin f. Let f and g be differentiable functions for all real numbers, what is f g h f g lim h 0 h? f g B. f g C. f g D. f g 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? Cubic B. Quartic C. Eponential D. Tangent E. none of these
4 NCMATYC Calculus Tournament Fall 0 6. Let f and g be differentiable functions such that g f is the value of g?, f 5, and f B. 8 C. 0 D. E What 7. What is lim 0? B. undefined C. D. 0 E. none of these 8. Determine d y d for y. d y y d B. d y y d y C. d y y d 9y D. d y d 9 E. d y d 9y 9. Compute ln d ln C ln C B. ln C C. ln C D. ln C E. 0. Which of the following is an inflection point on the graph of y? ( ) 0 There is no inflection point B. 8, C., D. 8, E. none of these 9 5 7
5 NCMATYC Calculus Tournament Fall 0. What is the equation of the tangent line to the graph of ( ) ( ) y ( ) at 0? y7 B. y C. y D. y7 E. none of these. Evaluate: lim e B. e C. e D. 0 E.. For how many integer values of c does lim equal a real number? c B. 0 C. D. infinitely many E. impossible to determine. For how many values of the domain does the function f e cos have a horizontal tangent line? 0 B. C. D. E. infinitely many 5. Which of the following is the derivative of the function F f g h i. F f g f g h f g h ii. F g h gh f f g h iii. F f h f hg f gh? i B. ii C. iii D. none of these E. all of these
6 NCMATYC Calculus Tournament Fall 0 6. Evaluate 0 d B. C. D. 8 E. none of these 7. Evaluate the first derivative of y sin. dy sin d cos ln B. dy d sin sin dy sin d cos ln sin D. dy sin cos ln E. d dy d cos sin 8. Define h r dr. Evaluate h. h 6 9 B. h 6 9 C. h 9 D. h 9 E. h 9 9. Compute the local etrema of 5ln y for. ln ln No local etrema for B. these 0 e, ln e D. e,5 C. 5, ln 5 e e E. none of 0. Suppose the radius of a cylinder is measured at cm with a possible error of ±0.0 cm and the height is measured at 5 cm with a possible error of ± 0. cm. Estimate the error in the calculated volume using differentials to the nearest hundredth of a cubic centimeter..π cm B..π cm C. 0.6π cm D. 0.8π cm E..0π cm
7 NCMATYC Calculus Tournament Fall 0. Let the quadratic function f a b c be non-negative on the interval, following is f d? f f f f f f. Which of the f f f 0 B. 0 C. 0 f f f D. 0 E. None of these. Which of the following is represents the volume of the solid formed by revolving the region bounded by the graphs of y ln, -ais, and the line about the line. ln y 9 ln d B. 9 0 D. ln ln y C. 0 e dy d E. None of these e dy. The intersection of y and y creates two tangent lines, one to each curve. Which of the following epressions gives the angle created by the intersection of these two tangent lines? tan tan B. tan 8 tan C. tan tan D. tan 8 tan 8 E. tan tan. L be the tangent line to at the point, y y 5. What is the sum of the intercepts of L? 5 B. 6 9 C. 7 D. 5 6 E. 0 5
8 NCMATYC Calculus Tournament Fall 0 5. What is the arc length of y e e from 0 to ln? B. C. 8 D. 6 E. 6. Determine the minimal distance from the point 5,0 to the graph y. 5 B. 5 C. D. 6 E A circle centered at the origin passes through the point,. Determine the area in Quadrant IV between the tangent line to the circle at, and the circle. 6 5 B. 5 C. 0 D E Let R be the region in Quadrant I bounded by y, the line y 0, and the line 0. Calculate the volume of the solid of revolution generated by revolving R about the line. 9 0 B. 9 C. 5 D. 5 E What is the value of d? 0 B. C. D. 6 E. 8 6
9 NCMATYC Calculus Tournament Fall 0 0. Determine d y d if y y = y y y B. y C. y y D. y y E. None of these. 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?. min B. min C. min D..8 min E..6 min. Determine the area of the largest rectangle that can be inscribed in a semicircle of radius cm. 8 cm B. 6 cm C. cm D. 0 cm E. cm. Determine the equation of the normal line to the curve y ln at 0. y B. y C. y D. y E. y. Evaluate 0 cost sin e t dt. 6 e B. e C. e D. e E. e 5. What is the length of the curve sin y? B. C. D. E. none of these 7
10 NCMATYC Calculus Tournament Fall 0 6. Let dv tv t e dt t and v 0 8. What is vt? t t v t t e 8e B. t t v t te 8e C. t t v t t e 8e D. v t 8e t E. none of these 7. Evaluate n. n 9 n It is divergent. B. 5 C. 5 D. 8 E Which of the following is the general solution for the differential equation dy d y e? e e y d C B. y d C D. y e d C E. y e d C C. y e d C 9. A rope is 0 feet long and weighs 0.5 lb/ft. If it hangs over the edge of a building 00 feet tall, how much work is done by pulling half of the rope to the top of the building? 0 ft-lb B. 00 ft-lb C. 60 ft-lb D. 80 ft-lb E. none of these 0. Find the centroid of the region bounded by the curve + y + 0y + 5 = 0 (0, 0) B. (, -) C. (, -) D. (, -5) E. (, -6) 8
lim x c) lim 7. Using the guidelines discussed in class (domain, intercepts, symmetry, asymptotes, and sign analysis to
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