Arkansas Council of Teachers of Mathematics 2013 State Contest Calculus Exam

Transcription

1 0 State Contest Calculus Eam In each of the following choose the BEST answer and shade the corresponding letter on the Scantron Sheet. Answer all multiple choice questions before attempting the tie-breaker questions. The tie-breaker questions at the end are to be used to resolve any ties between st, nd, and/or rd place. Be sure that your name is printed on each of the tiebreaker pages. The figures are not necessarily drawn to scale. You may use a scientific graphing calculator such as a TI-8. Good Luck!. Let, where and are the functions whose graphs are shown. Find A. B. C. D. E.. Let, where and are the functions whose graphs are shown. Find A. 6 B. 6 C. 0 D. E.

2 0 State Contest - Calculus. Suppose and for all values of. How large can possibly be? A. 0 B. C. D. E. b. ' a f d A. f b f a B. f ' b f ' a C. f a f b D. f ' a f ' b E. f ' a f ' b 5. A particle moves along the curve. As it reaches the point, the -coordinate is increasing at a rate of. How fast is the -coordinate of the point changing at that instant? A. cm / s B. 6 cm / s C. cm / s D. cm / s E. 8 cm / s 6. The figure shows the graphs of three functions. One is the position function of a car, one is the velocity function of the car, and one is the acceleration function of the car. Determine the valid statement. A. is the position function. B. is the velocity function. C. is the acceleration function. D. b is the position function. E. is the velocity function.

3 0 State Contest - Calculus 7. A canister is dropped from a helicopter at a height of 90 m. Its parachute does not open, but it is designed to withstand an impact velocity of 00 m/s. If we use 9.8 m/s for the acceleration due to gravity and ignore wind resistance, then how fast is it going when it hits the ground? A. 0 m/s B. 98 m/s C. 9.8 m/s D. 980 m/s E. 0 m/s 8. A trough is feet long and its cross-section has the shape of an isosceles triangle that is feet at the top and has a height of foot. If the trough is being filled with water at a rate of ft /min, then how fast is the water level rising when the water is 6 inches deep? A. ft / min B. / min ft C. 6 ft / min D. / min 6 ft E. / min 9 ft 9. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths and if the two sides of the rectangle lie along the legs. ab A. B. ab C. ab D. ab ab E. 0. Suppose that we approimate a definite integral with a Left Riemann Sum. Which of the following statements is true? A. The estimate is lower than the actual integral if the function is increasing on the interval. B. The estimate is lower than the actual integral if the function is decreasing on the interval. C. The estimate is lower than the actual integral if the function is concave up on the interval. D. The estimate is lower than the actual integral if the function is concave down on the interval.

4 0 State Contest - Calculus. Evaluate the limit by recognizing the epression as a limit of a Riemann sum for a function defined on and applying the Fundamental Theorem of Calculus. A. 5 B. 9 C. D. E.. Suppose that we approimate a definite integral with the Trapezoidal Rule. Which of the following statements is true? A. The estimate is lower than the actual integral if the function is increasing on the interval. B. The estimate is lower than the actual integral if the function is decreasing on the interval. C. The estimate is lower than the actual integral if the function is concave up on the interval. D. The estimate is lower than the actual integral if the function is concave down on the interval.. Approimate 9 5 d whole number. A. 0,5 B. 9,700 C. 88,57 D. 99,00 using the Trapezoidal Rule with intervals. Round your answer to the nearest d.. Evaluate the following: sin A. cos C B. cos C C. cos C D.

5 0 State Contest - Calculus 5. 6 lim. By the definition of a limit, there is a positive real number such that A. 0.0 B. 0.0 C. 0. D. 0. E Compute A. B. C. D. tan6 d tan d e e tan e sec 6 sec 6e 6e sec tan if 0. The largest valid value of is 7. lim 0 sin A. B. C. D. 0 f. 8. Find and simplify a formula for the difference quotient where A. B. C. D. 9. Let f cos A. 0 B. C. - D. -65,56 E. 65,56. f 6 0 5

6 0 State Contest - Calculus 0. The probability of an event [a, b] for a continuous probability distribution is the area between the graph of the probability density function (pdf) for the distribution and the -ais over that interval. The wait time for starting service at a checkout line has probability distribution e if 0 pdf. 0 otherwise ( is measured in minutes.) What is the probability that the wait time will be between and minutes? A. 0.5 B. 0. C. 0.8 D. 0. E The mean of a continuous probability distribution is E wait time for the distribution from the previous question? A..0 minutes B..0 minutes C. 5. minutes D.. minutes pdf d. What is the mean For questions - let R be the region bounded by the curves y sin and y.. Approimate the area of the region R. Round your answer to four decimal places. A. 0.7 B. 0. C D..5. Approimate the volume of the solid formed by revolving the region R about the -ais. Round your answer to four decimal places. A..556 B C..808 D Approimate the volume of the solid formed by revolving the region R about the y-ais. Round your answer to four decimal places. A..556 B C..808 D

7 0 State Contest - Calculus Tie-Breaker Questions Name [Please Print] School [Please Print] In each of the following you must show supporting work for your answers to receive credit. The questions will be used in the order given to resolve ties for st, nd, and/or rd place. Be sure that your name is printed on each of the tiebreaker pages.. For a continuous probability distribution the probability that the random variable is in the interval [a, b] is the area bounded by the -ais, = a, = b and the graph of the probability density function (pdf). The cumulative density function (cdf) for the distribution gives as its output the cumulative probability, i.e. cdf(k) = P( k). For a particular probability distribution we have 8 7,7 pdf 0 otherwise Find the formula for cdf(). Graph y = cdf()

8 0 State Contest - Calculus Tie-Breaker Questions Name [Please Print] School [Please Print]. State the Sum Rule for calculating the derivative of a sum of two differentiable functions f and g: d d f g Prove this result. 8

9 0 State Contest - Calculus Tie-Breaker Questions Name [Please Print] School [Please Print]. The rate of change of the number of coyotes N(t) in a population is directly proportional to 650-N(t), where t is the time in years. When t = 0 the population is 00, and when t = the population has increased to 500. Find the population when t =. 9

10 0 State Contest - Calculus Answers:. A. C. D. A 5. A 6. E 7. B 8. B 9. E 0. A. E. D. D. D 5. B 6. D 7. A 8. D 9. E 0. C. A. C. B. D 0

11 0 State Contest - Calculus Tie Breaker # Name Key For a continuous probability distribution, the probability that the random variable is in the interval [a, b] is the area bounded by the -ais, = a, = b and the graph of the probability density function (pdf). The cumulative density function (cdf) for the distribution gives as its output the cumulative probability, i.e. cdf(k) = P( k). For a particular probability distribution we have 8 7,7 pdf 0 otherwise Find the formula for cdf(). We see from the eplanation above that in general the cdf is an antiderivative of the pdf. Specifically, cdf pdf t dt. Notice that for < cdf 0dt 0 and note that For [, 7] 7 cdf t dt t t 7t t OR for > 7. pdf dt 0 dt cdf t dt u t du dt t t (Any of the last four versions on the left are good in this interval and can be used for the middle piece below.) 0 cdf 6,7 Graph y = cdf()

12 0 State Contest - Calculus Tie Breaker # Name Key State the Sum Rule for calculating the derivative of a sum of two differentiable functions f and g: ' ' d d f g f g Prove this result. d d f g f h g h f g lim h0 h f h g h f g lim h0 h f h f g h g lim h0 h f h f g h g lim h0 h f h f g h g lim h0 h h f h f g h g lim lim h0 h h0 h f ' g ' Definition of the derivative. Distributive Property Commutative Property Associative Property Distributive Property Limit of a sum is the sum of limits if they eist. Definition of a derivative.

13 0 State Contest - Calculus Tie Breaker # Name The rate of change of the number of coyotes N(t) in a population is directly proportional to 650-N(t), where t is the time in years. When t = 0 the population is 00, and when t = the population has increased to 500. Find the population when t =. dn k 650 N 650 N dn k dt dt ln 650 N kt C ln 650 N kt C 650 N e C 650 N e e 650 N C e N 650 C e t 0 N Ce C C 50 N e k0 t 0 N e 50 50e k e 7 k ln k ln 7 7 ktc kt kt kt kt k k ln 7t N e t 7 ln 7 N() e When t = the population is 55.