MATH 151, SPRING 2013 COMMON EXAM II - VERSION A. Print name (LAST, First): SECTION #: INSTRUCTOR: SEAT #:

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1 MATH 151, SPRING 2013 COMMON EXAM II - VERSION A Print name (LAST, First): SECTION #: INSTRUCTOR: SEAT #: THE AGGIE CODE OF HONOR "An Aggie does not lie, cheat, or steal, or tolerate those who do." By signing below, you indicate that all work is your own and that you have neither given nor received help from any external sources. SIGNATURE: PART I-MULTIPLE CHOICE The use of a calculator, laptop, or computer is prohibited. Mark the correct choice on your ScanTron using a No. 2 pencil. For your own records, also record your choices on your exam! Be sure to write your name, section and version letter of the exam on the ScanTron form. Each problem is worth 3 points. 1. A particle moves according to the equation s = t 2 t, t 0, where t is measured in seconds and s is in feet. What is the total distance the particle travels during the first 2 seconds? (a) 2 feet (b) 1 foot (c) 1 2 foot (d) feet (e) None of these 2. If y 3 x 2 = 1 7y, find dy dx. (a) 2x 3y (b) 1 + 2x 3y x 3y2 (c) x 3y2 (d) 7 (e) None of these 3. Which of the following is the 2013th derivative of f(x) = cos x? (a) None of these (b) sin x (c) sin x (d) cos x (e) cos x

2 For questions 4-6, suppose f and g are differentiable functions whose values at x = 1 are given in the table below: f(1) = 2 g(1) = 1 f (1) = 3 g (1) = 2 4. If U(x) = g(x) f(x), what is U (1)? (a) 7 (b) None of these (c) 7 4 (d) 2 3 (e) If V (x) = g(x)f(x), what is V (1)? (a) None of these (b) 7 (c) 6 (d) 1 (e) 7 6. If S(x) = (f(x)) 4, what is S (1)? (a) 108 (b) 96 (c) 24 (d) None of these (e) Using the linear or differential approximation to f(x) = 1 x to 1 2.1? (a) (b) (c) 20 (d) None of these (e) at a = 2, which of the following is an approximation

3 8. Given the curve parametrized by x = 2t 3, y = sin(πt), find the slope of the line tangent to the curve at the point (2, 0). (a) None of these (b) 24 π (c) 6 π π (d) 24 (e) π 6 9. A dog on a chain sits 1 meter from a telephone pole. A squirrel begins running up the pole at a speed of 1 2 meter per second. How fast is the distance changing (in m/s) between the squirrel and the spot the dog is stting on when the squirrel has traveled 3 meters up the pole? (a) None of these (b) (c) 2 10 (d) 3 3 (e) If f (15) (0) = 2 and H(x) = f (a) 2 14 (b) None of these 1 (c) 2 14 (d) 2 15 (e) ( x 2 ), find H (15) (0). 11. A model for predicting the earth s population growth is given by P (t) = is in millions of people. What is lim t P (t)? (a) 5 (b) (c) 500 (d) 100 (e) None of these 500, where t is in years and P e 2t

4 12. Which of the following is true about the line tangent to x = t 2 2t + 1, y = t 3 6t at the point (0, 4)? (a) The tangent line is vertical (b) The tangent line has negative slope (c) The tangent line is horizontal (d) None of these (e) The tangent line has positive slope 13. Which of the following is d ( cos(sin x) )? dx (a) cos(cos x) sin x (b) cos(cos x) sin(sin x) (c) None of these (d) sin(cos x) (e) sin(sin x) cos x 14. Which of the following is a valid explanation for lim x 0 sin(5x) sin(7x) = 5 7? (a) sin(5 0) sin(7 0) = 5 7 (b) sin(5x) sin(7x) = 5 sin x 7 sin x 5 7 (c) sin(5x) sin(7x) = sin(5x) 7x 5x sin(7x) 5x 7x 5 7 (d) None of these (e) sin(5x) sin(7x) = 5 cos(5x) 7 cos(7x) Find a tangent vector of unit length to the curve r(t) = (t 2 + 3t)i 1 j at the point corresponding to t = 1. t2 (a) 4 17 i 1 17 j (b) None of these 10 (c) i 1 j (d) (e) 5 i + 2 j i 1 26 j

5 PART II 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. Given f(x) = x x : (a) (7 pts) Find an equation of the line tangent to f at the point where x = 2. (b) (3 pts) Find the points (x-coordinates only) on the curve where the tangent line is horizontal.

6 17. (6 pts) Given y = 1 5 xe4x, find and completely simplify y 3y 4y. 18. Given the equation x 3 x 2 y 2 + 3y 1 = 0 : (a) (7 pts) Find dy dx. (b) (3 pts) Find the equation of the line tangent to the curve at the point ( 1, 1).

7 19. Let C denote the curve given by the vector equation r(t) = e t2 +t, 1 6t (where < t < 1 6 ). (a) (4 pts) Find r (t). (b) (4 pts) If an object moves along the curve C according to the equation above, find the speed of the object at the point corresponding to t = 0. (c) (3 pts) Find a vector equation of the line tangent to C at the point corresponding to t = 0.

8 20. Suppose F and G are differentiable functions. The line y = 1 + 2x is the linear (tangent-line) approximation to F at x = 2, and the line y = 2 3x is the linear (tangent-line) approximation to G at x = 2. (a) (4 pts) Find F (2), F (2), G(2), and G (2). (b) (5 pts) Let H(x) = F (x). Find the linear (tangent-line) approximation to H at x = 2. G(x)

9 21. (9 pts) A spectator watches a helicopter which flies at a constant altitude of 80 meters at a speed of 10 m/s. At what rate is the angle of elevation from the feet of the spectator to the helicopter (see figure below) changing 6 seconds after the helicopter passes directly above the spectator?. θ. SCORE SHEET: DO NOT WRITE BELOW! Question TOTAL Max Pts Score.

10 MATH 151 Exam II Print name (LAST, First): SECTION #:

MATH 151, FALL 2017 COMMON EXAM 2, VERSION B. LAST NAME (print) : FIRST NAME (print): INSTRUCTOR : SECTION NUMBER: DIRECTIONS THE AGGIE HONOR CODE

MATH 151, FALL 2017 COMMON EXAM 2, VERSION B. LAST NAME (print) : FIRST NAME (print): INSTRUCTOR : SECTION NUMBER: DIRECTIONS THE AGGIE HONOR CODE MATH 5, FALL 7 COMMON EXAM, VERSION B LAST NAME (print) : FIRST NAME (print): INSTRUCTOR : SECTION NUMBER: DIRECTIONS. The use of a calculator, laptop, or computer is prohibited.. TURN OFF cell phones