UNIVERSITY OF REGINA Department of Mathematics and Statistics. Calculus I Mathematics 110. Final Exam, Winter 2013 (April 25 th )

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1 UNIVERSITY OF REGINA Department of Mathematics and Statistics Calculus I Mathematics 110 Final Exam, Winter 2013 (April 25 th ) Time: 3 hours Pages: 11 Full Name: Student Number: Instructor: (check one) Shahla Nasserasr ( ) Robert Petry (100-C01) INSTRUCTIONS 1. All work and answers are to be placed on this exam. If you require more space for an answer, work on the back of the page or on the extra blank page at the end of the exam. In either case indicate on the question where further work is to be found. 2. To receive full credit for correct answers it is necessary to show all of your work. 3. Scrap paper is provided for rough work only and should not be submitted for grading. 4. Good luck! For instructor use only: Question: Total Marks: Score:

2 MATH 110 Final Winter 2013 Page 2 Student Number: 1. (12 marks) Evaluate each of the following limits. x 2 (a) lim x 2 x 2 2x (b) lim x x2 + 3 x + 2 (c) lim t 0 t t 2 (d) lim x 0 tan x x

3 MATH 110 Final Winter 2013 Page 3 Student Number: 2. (5 marks) Do one of the following three problems. Circle the letter of the question you have chosen to answer. (a) Determine whether the following function f is continuous at x = 1: f(x) = (x 2) 2 x if x < 1 x if x 1 (b) Use the Intermediate Value Theorem to show that the equation 3 x = 1 x has at least one solution in the interval (0, 1). (c) Verify that the function f(x) = x 3 8x 5 satisfies the hypotheses of the Mean Value Theorem on the interval [1, 4]. Then find a number c in the open interval (1, 4) that satisfies the conclusion of the theorem.

4 MATH 110 Final Winter 2013 Page 4 Student Number: 3. (15 marks) Differentiate the following functions. You do not need to simplify your answers. (a) f(x) = x 6 2x 3 + x 3 (b) f(x) = 3 x(1 + cos x) 10 (c) f(x) = sin x tan x (d) f(x) = x (x 1) 2

5 MATH 110 Final Winter 2013 Page 5 Student Number: 4. (5 marks) Using implicit differentiation, find the derivative dy dx if y cos x = x 2 + y (5 marks) Find the equation of the tangent line to the curve y = 2 at the point (2, 1). x

6 MATH 110 Final Winter 2013 Page 6 Student Number: 6. (9 marks) Find the absolute maximum and absolute minimum values of the function f(x) = 2x 3 + 3x 2 36x over the closed interval [ 1, 4].

7 MATH 110 Final Winter 2013 Page 7 Student Number: 7. (10 marks) Do one of the following two problems. Circle the letter of the question you have chosen to answer. (a) A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 2 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? (b) A mirror is to be made in the shape of a rectangle with an equilateral triangle on top as shown. The mirror is to have a perimeter of 1 metre. What should the width w and the height h of the rectangle be if the area of the mirror is to be maximized? (You may use that the area of an equilateral triangle with side length w is A = 3 4 w2.) h w w w

8 MATH 110 Final Winter 2013 Page 8 Student Number: 8. (13 marks) Find intercepts, asymptotes, local maxima and minima, and inflection points, if they exist, for the graph of f(x) = x2 4 and then sketch the graph. You may x 2 1 assume that f 6x (x) = and f (x) = 18x2 + 6 (x 2 1) 2 (x 2 1). 3 (Use the back of this page if you need extra room for your sketch.)

9 MATH 110 Final Winter 2013 Page 9 Student Number: 9. (16 marks) Evaluate each of the following indefinite and definite integrals. (a) ( x + x 3 ) dx (b) 0 2 x x 4 dx (c) x x 2 dx (d) sin x cos 4 x dx

10 MATH 110 Final Winter 2013 Page 10 Student Number: 10. (10 marks) Find the area of the region R bounded by the graphs of the functions y = 2x and y = x 2 2x.

11 MATH 110 Final Winter 2013 Page 11 Student Number: This page is intentionally left blank. If you use it for work that is to be graded, please indicate on the page of the question that the work is to be found here.