MATH 151, SPRING 2018

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1 MATH 151, SPRING 2018 COMMON EXAM II - VERSIONBKEY 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 1-20), 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!. In Part 2 (Problems 21-2), 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.. Again. The use of a calculator, laptop or computer is prohibited. THE AGGIE HONOR CODE An Aggie does not lie, cheat, or steal, or tolerate those who do. Signature: 1

2 FOR INSTRUCTOR USE ONLY Question Points Awarded Points TOTAL 100 2

3 Part 1: Multiple Choice (3 points each) 1. Find h (x) for h(x) = e sinx. (a) h (x) = e sinx (b) h (x) = cosx e cosx +sin 2 x e cosx (c) h (x) = sinx cosx e sinx (d) h (x) = cosx e sinx (e) h (x) = sinx e sinx +cos 2 x e sinx key B 2. Find the value of x where the tangent line to the graph of f(x) = x 1 3 is parallel to the line 2x 12y = 9. (a) 1 5 (b) 1 3 (c) 1 (d) 2 key B (e) 5 3. Find a tangent vector to the curve r(t) = e 3t 2,(2t+1) 3 at the point where t = 1. (a) e,27 (b) e,5 (c) 3e,9 (d) 3e,27 (e) 3e,5 key B. For what value(s) of t does the graph of x = t 3 12t, y = 2t 3 +t 2 have a vertical tangent? (a) t = 1,1 (b) t = 0, 2 3,2 3 (c) t = 0,12 (d) t = 2,2 key B (e) There is no vertical tangent. 3

4 ( (x+5) 5. Find f (x) for f(x) = ln ). [Hint: First use properties of logarithms.] x 7 (a) x+5 3 x 7 1 (b) (x+5) 1 x 7 (c) (x 7) 3(x+5) (d) x+5 1 2(x 7) (e) x+5 2 x 7 key B. The length of a rectangle is increasing at a rate of cm/s and its width is decreasing at a rate of 5 cm/s. When the length is 12 cm and the width is 8 cm, how fast (in cm 2 /s) is the area of the changing at that moment? (a) 28 key B (b) 8 (c) 92 (d) 8 (e) Find the slope of the tangent line to the curve parametrized by x = t 2 +e 2t, y = t+t 2 at the point (1,0). (a) 0 (b) 1 2 key B 3 (c) 2+2e 2 (d) 2 (e) The slope is undefined.

5 8. A ball is tossed in the air, and the height of the ball at time t seconds is given by h(t) = 2t t 2, where h(t) is measured in feet. What is the maximum height of the ball? (a) 3 ft (b) 18 ft (c) 3 ft key B (d) ft (e) 72 ft 9. Find the slope of the tangent line to the graph of f(x) = (lnx) 2 at x = e 3. (a) e (b) e 3 key B (c) 12 (d) (e) e 2 2 e 3 2 e 10. Find f (x) if f(x) = cos(sinx). (a) sin(sinx) cosx sinx 2 cos(sinx) (b) sin2 x cosx 2 cos(sinx) (c) sin(sinx) cosx 2 key B cos(sinx) cosx (d) 2 cos(sinx) (e) sinx cosx 2 cos(sinx) 5

6 11. A bacteria culture stars with 3 million bacteria and the population triples every 30 minutes. Find the number of bacteria after 90 minutes. (a) 2 million (b) 27 million (c) 5 million (d) 81 million key B (e) 108 million 12. Use the linear approximation of f(x) = x at x = 2 to find an approximate value of (2.01). (a) 1.12 (b) 1.2 (c) 1.32 key B (d) 1.8 (e) Find f (2018) (x) for f(x) = xe x. (a) xe x (b) 2018xe x (c) xe 2018x (d) (2018 x)e x key B (e) (x 2018)e x

7 1. A particle moves according to the equation of motion s(t) = t 2 t + 5 where s(t) is measured in feet and t is measured in seconds. Find the total distance traveled in the first 3 seconds. (a) 0 ft (b) 2 ft (c) 3 ft (d) 5 ft key B (e) 9 ft 15. An object is moving according to the equation of motion s(t) = sint+ 1 t2. Find the time(s) when the acceleration is zero for 0 t 2π. (a) t = π, 11π (b) t = π 3, 5π 3 (c) t = 5π, 7π (d) t = 7π, 11π (e) t = π, 5π key B 1. Find the values of a and b so that the tangent line to the parabola y = ax 2 +b at x = 1 has equation y = 5x 9. (a) a = 5 2, b = 13 2 key B (b) a = 5 2, b = 3 2 (c) a = 1 2, b = 13 2 (d) a = 1 2, b = 13 2 (e) a = 9, b = 13 7

8 (17-18) Suppose f and g are differentiable functions which satisfy the following condition. 17. Let u(x) = f(x) g(x). Find u (1). (a) 0 (b) 1 (c) 3 (d) 5 key B (e) Not enough information to be determined. x f(x) f (x) g(x) g (x) Let v(x) = f(g(x)). Find v ( 1). (a) -2 (b) -1 (c) 0 (d) 5 (e) Not enough information to be determined. key B 19. Find all the values of x on the interval [0,2π] for which the tangent line to the graph of f(x) = cos 2 x + sinx is horizontal. (a) x = π 2, 3π 2, π, 5π key B (b) x = π 2, 3π 2, 7π, 11π (c) x = π 2, 3π 2 (d) x = π 2, 3π 2, π 3, 5π 3 (e) x = π 2, 3π 2, 2π 3, π Suppose the linear approximation for the function f(x) at a = 9 is given by y = 2x 2. If g(x) = f(x), find the linear approximation for g(x) at a = 9. (a) L(x) = +2(x 9) (b) L(x) = + 1 (x 9) key B (c) L(x) = 2+(x 9) (d) L(x) = (x 9) (e) L(x) = (x 9) 8

9 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. 21. (8 pts) For the equation y = e 2x +e 3x, find y +y y. key B: (10 pts) Find dy dx for the equation sin(3xy2 )+5e y = 3x 2 y. key B: dy dx = xy 3y 2 cos(3xy 2 ) xycos(3xy 2 )+5e y 3x 2 9

10 23. (10 pts) Water is leaking out of an inverted conical tank at a rate of ft 3 /min. The tank has height 7 ft and the radius at the top is ft. Find the rate at which the water level is decreasing when the water is 2 ft high. (The volume of a cone is V = 1 3 πr2 h.) key B: 9 1π ft/min 10

11 2. (12 pts) Find f (x). (a) f(x) = sec 2 (3x)+arcsin(x). key B: f (x) = sec(3x) sec(3x) tan(3x)+ 1 (x) 2 (b) f(x) = (tanx) (x3). key B: f (x) = (tanx) (x3 ) ( ) 3x 2 ln(tanx)+x 3 sec2 x tanx 11

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