Arkansas Council of Teachers of Mathematics 2012 State Competition Calculus Exam

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1 Arkansas Council of Teachers of Mathematics 2012 State Competition Calculus Exam ACTM State Calculus Competition Spring 2012 Page 1 For questions 1 through 25, mark your answer choice on the answer sheet provide Unless otherwise stated, assume all variables are real and all functions are continuous over relevant domains. Assume all angles are in radians. After completing items 1 through 25, answer each of the tiebreaker items in sequential order (do #1 first, followed by #2, and then #3 last). Be sure that your name is printed on each of the tiebreaker pages. Congratulations for being selected to participate in the ACTM State Contest! 1. Consider the following limit: Given, determine the value for to satisfy the definition of the limit. 25 Cannot be determine 2. Determine the slope of the normal line to the function ( ) at the point. 3. The US Department of Energy Statistics found that the rate of consumption of gasoline in the US is approximated by with corresponding to 1980, and is in millions of gallons. Determine the number of gallons consumed between 1980 and Round your answer to the nearest one hundredth million gallons 2.74 million gallons million gallons 39, million gallons. Cannot be determined

2 4. The equation of the tangent line(s) to the curve y x 1 4 y 1 4 4x 16y 16 a, b and c none of these 5. Determine by differentiating. Do t s p fy ACTM State Calculus Competition Spring 2012 Page 2 2 x y at x = 4 is ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6. Determine all values of x where the function and its derivative intersect. No intersection points 7. Find the global maximum to the following function on the given domain. o t t There is no maximum Given f ( x) x 4x 4x on the interval [-1, 3], then the value(s) of c such that f (3) f ( 1) f ( c) is/are 3 ( 1) c = 0 c = 1 c = 2 all of these none of these

3 ACTM State Calculus Competition Spring 2012 Page 3 9. Determine the minimum number of critical numbers that a cubic function of the form may hav Find all values of where the following function is continuous. 2 x 2 2 x Evaluate the following integral s s os (s os) (s os) os 12. The values of a and b, respectively, which make the function both continuous and differentiable at x = 1 are 3 2 x ax 1, x 1 f ( x) 2 x bx 2, x 1-4 and -3-1 and 0-1 and 1-3 and -1 None of these 13. What is/are the region(s) where the following function is concave down on the domain? s The function has no regions where it is concave down.

4 ACTM State Calculus Competition Spring 2012 Page Solve the differential equation with the given initial conditions. Cannot be determined 15. The relationship between the temperature F on the Fahrenheit scale and the temperature C on the Celsius scale is given by. Find the Rate of Change of F with respect to C. 16. Calculate the volume of the solid of revolution generated by the region bounded by the curve and the lines x = 0 and, rotated around the y axis. Use the disk metho o of t s 17. Determine the sum of the following: The summation has no value 18. Ro s T o pp s to t fu t o on the interval. Find the value(s) of that satisfy the conclusion of Ro s T o

5 ACTM State Calculus Competition Spring 2012 Page Estimate the area under the curve bounded by, the -axis, and the -axis, using four rectangles of equal width. Use right hand end points. 20. Evaluate the following limit: the limit does not exist lim x 3 x 3 4x 2 21x x Determine the range of the following function: { } { } { } { } y y.25, y Use implicit differentiation to find the slope of the tangent line to the function at the indicated point. s 23. Use the Fundamental Theorem of Calculus to evaluate the following Cannot be evaluated

6 ACTM State Calculus Competition Spring 2012 Page Consider the function, s { How many vertical asymptotes does this function have? It has no vertical asymptotes. 25. Apply the Mean Value Theorem for Definite Integrals to the following function. o t t Determine the value z, such that None of the above

7 ACTM State Calculus Competition Spring 2012 Page 7 Name School Tiebreaker Questions Your solutions should be written clearly. All work leading to your final answer must be include The questions will be used in sequential order to resolve ties for first, second, and/or third plac Tiebreaker #1. Solve the following integration problem.

8 ACTM State Calculus Competition Spring 2012 Page 8 Name School Tiebreaker #2. If f x ( x) ( x) x. Find (x) f.

9 ACTM State Calculus Competition Spring 2012 Page 9 Name School Tiebreaker #3. The Gateway Arch in St. Louis has the shape of an inverted catenary. The shape is approximated by the function os where the hyperbolic cosine function, e cosh( x) x e 2 x Approximate the total open area under the arch.

10 Arkansas Council of Teachers of Mathematics 2012 State Competition Calculus Exam Solutions ACTM State Calculus Competition Spring 2012 Page 10 Multiple Choice Answers 1....B 2....D 3....C 4....D 5....E 6....B 7....C 8....D 9....A D D A D B B C C B D E E A B C C

11 ACTM State Calculus Competition Spring 2012 Page 11 Tie Breaker Question 1 Solution 1. Solve the following integration problem. t f st g t s o s t ook so b Us g o g s o u s t p ob to so t g manageabl By long division, Thus, Tie Breaker Question 2 Solution 2. If f x ( x) ( x) x. Find (x) f. f x 1 2 x x x *(ln( x)) x *ln( x) 1 x x, x >0 Tie Breaker Question 3 Solution 3. The Gateway Arch in St. Louis has the shape of an inverted catenary. The shape is approximated by the function os where cosh is the hyperbolic cosine function. Approximate the total open area under the arch. The key to this problem is.

ACTM Regional Calculus Competition March 3, 2012

Regional Calculus Test Spring 2012 Page 1 ACTM Regional Calculus Competition March 3, 2012 Instructions: Select the best choice for each question. Afterward, attempt the tie-breaker questions in sequential