MATH 151, FALL 2017 COMMON EXAM III - VERSION B

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1 MATH 151, FALL 2017 COMMON EXAM III - VERSION B LAST NAME(print): FIRST NAME(print): INSTRUCTOR: SECTION NUMBER: DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your exam will be collected and you will receive a zero. 3. In Part 1 (Problems 1-15), mark the correct choice on your ScanTron using a No. 2 pencil. The scantrons will not be returned, therefore for your own records, also record your choices on your exam! 4. In Part 2 (Problems 16-19), present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 5. Be sure to write your name, section number and version letter of the exam on the ScanTron form. THE AGGIE HONOR CODE An Aggie does not lie, cheat, or steal, or tolerate those who do. Signature: 1

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3 1. Given that (a) 19 (b) 11 (c) 7 (d) 15 1 f(x) dx = 3, 2 Part 1: Multiple Choice (4 points each) f(x) dx = 2 and 0 g(x) dx = 4, determine the value of (e) The value of the integral cannot be determined. 2 (3f(x) + g(x)) dx. 2. Find f(1) if f (x) = e x + cos x 2x and f(0) = 5. (a) e + sin (b) e + sin (c) e sin (d) e sin (e) e sin Given that the second derivative of a function f with domain all real numbers is f (x) = e x (4 x)(2+x), determine the interval(s) where f is concave down. (a) ( 2, 4) (b) ( 2, 0), (4, ) (c) (, 2), (0, 4) (d) (, 2), (4, ) (e) (, 0) 4. Which of the following gives the exact area under the curve f(x) = sin x on the interval [1, 3]. ( ) 3 3i (a) lim n n sin n ( ) 2 2i (b) lim n n sin n ( 3 (c) lim n n sin 1 + 3i ) n ( 2 (d) lim n n sin 1 + 2i ) n ( 2 (e) lim n n sin 1 + 3i ) n 3

4 5. Which of the following statements is true concerning the local extrema of f(x) = x ln x? (a) f has a local maximum at x = 1 e. (b) f has a local minimum at x = 1 e. (c) f has a local minimum at x = 0 and a local maximum at x = 1 e. (d) f has a local maximum at x = 0 and a local minimum at x = 1 e. (e) f has no local extrema. 6. Use the graph of f(x) below to evaluate (a) 19/2 (b) 1/2 (c) 11/2 (d) 7/2 (e) 9/2 5 2 f(x) dx. ( ) 3 7. Evaluate lim x tan. x x (a) 0 (b) (c) 3 (d) (e) 3 8. Approximate the area under the curve f(x) = x on the interval [1, 7] using 3 equal width-rectangles and using left endpoints. (a) 76 (b) 38 (c) 172 (d) 86 (e) 118 4

5 9. Find all critical numbers of the function f(x) = x 2/3 (x + 1) 2. (a) x = 1, 0 (b) x = 1, 1 4 (c) x = 1, 1 2, 0 (d) x = 1, 1 2 (e) x = 1, 1 4, Evaluate lim x 0 e x + e x 2 x 2. (a) 1 2 (b) 0 (c) 1 (d) 1 (e) Find the absolute maximum and minimum values of the function f(x) = x x (a) Absolute Maximum of 10 ; Absolute Minimum of 0 (b) Absolute Maximum of 1 2 ; Absolute Minimum of (c) Absolute Maximum of 10 ; Absolute Minimum of 1 2 (d) Absolute Maximum of 0; Absolute Minimum of 1 2 (e) Absolute Maximum of 1 2 ; Absolute Minimum of 0 on the interval [0, 3]. 5

6 12. Find the intervals where f(x) = 3x 5 5x 3 is increasing. (a) ( 1, 0), (0, 1) (b) ( 1, 0), (1, ) (c) (, 1), (1, ) (d) (, 1), (0, 1) (e) (, 1), (0, ) 13. Find the value(s) of c that satisfy the conclusion of the Mean Value Theorem for the function f(x) = x 2 5x + 3 on the interval [1, 3]. (a) c = 2 (b) c = 1 (c) c = 1 (d) c = 3 (e) The Mean Value Theorem is not valid for this function on the given interval. The graph below is the graph of f (x), the first derivative of a function f. Use this graph to answer Questions For what value of x does f have a local minimum? (a) x = 0 (b) x = 4 (c) x = 3 (d) x = 2 (e) f does not have any local minima. 15. For what value(s) of x does f have an inflection point? (a) x = 0 and x = 3 (b) x = 0 and x = 2 (c) x = 0 and x = 4 (d) x = 3 only (e) x = 2 only 6

7 Part 2: Work Out Directions: Present your solutions in the space provided. Show all your work neatly and concisely and box your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 16. (10 points) The acceleration of a particle is given by a(t) = 2 t + 3, for t > 0. Find the position function s(t) for the particle if the velocity and position at time t = 1 are v(1) = 5 and s(1) = 4, respectively. t (10 points) Calculate lim x 0 (cos 5x) 1/x2. 7

8 18. (10 points) Consider the function f(x) = x 4 x 2. (a) What is the domain of f? (b) Determine the interval(s) where f is increasing and decreasing and any local extrema. f is increasing on f is decreasing on f has local maxima at x = f has local minima at x = (c) Determine the interval(s) where f is concave up and concave down. f is concave up on f is concave down on 8

9 19. (10 points) A box with a square base and an open top is being constructed from 2700 m 2 of material. Determine the side length of the base that will maximize the volume of the box. Briefly show that your answer gives a maximum. h x x FOR INSTRUCTOR USE ONLY Question Points Awarded Points TOTAL 100 9