MATH 121 FINAL EXAM FALL December 6, 2010


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1 MATH 11 FINAL EXAM FALL December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic devices are allowed. If your section number is missing or incorrect, 5 points will be deducted from the total score. You may only use techniques that were discussed in class. Simplify all of your answers. Points PAGE 1 4 points Score PAGE 1 points PAGE 1 points PAGE 4 18 points PAGE 5 16 points Points PAGE 6 16 points Score PAGE 7 7 points PAGE 8 10 points PAGES points Raw Score (out of 15): Final Score (100 * Raw Score / 15 ):
2 Final Exam MATH 11 Fall (6 points each) Compute dy dx. a. x x y 5x x x e b. 5 1 y x tan x 4 c. y cos x d. y x 1 x 1
3 Final Exam MATH 11 Fall x. (6 points) Find all values of x at which the tangent line to the curve y x 9 is horizontal.. (6 points) Suppose that y is an implicit function of x and that of x and y. dy dx x y. Express d y dx in terms
4 Final Exam MATH 11 Fall ( points each) During the first 10 seconds of a rocket flight, the rocket is propelled straight up 1 so that in t seconds it reaches a height of st () t feet. a. What is the average velocity of the rocket during the first 10 seconds of its flight? b. What is the instantaneous velocity of the rocket at t 10 seconds? 5. (6 points) (Version #1) Find values of the constants k and m that will make the following function continuous everywhere. x 5, x f( x) m( x1) k, 1 x x x7, x1
5 Final Exam MATH 11 Fall ( points each) During the first 10 seconds of a rocket flight, the rocket is propelled straight up 1 so that in t seconds it reaches a height of st () t feet. a. What is the average velocity of the rocket during the first 10 seconds of its flight? b. What is the instantaneous velocity of the rocket at t 10 seconds? 5. (6 points) (Version #) Find values of the constants k and m that will make the following function continuous everywhere. x x5, x1 f ( x) mk( x), x1 x 7, x
6 Final Exam MATH 11 Fall (6 points each) Compute the limits. a. x 9 lim x x b. lim x 4x x c. lim x x1 1 1 x 4
7 Final Exam MATH 11 Fall (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1 cm min. At what rate is the radius of the snowball changing when the radius is 5 cm? Recall that the surface area S of a sphere with radius r is S 4 r. 8. (8 points) (Version #1) Use an appropriate local linear approximation to estimate.0. 5
8 Final Exam MATH 11 Fall (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1 cm min. At what rate is the radius of the snowball changing when the radius is 5 cm? Recall that the surface area S of a sphere with radius r is S 4 r. 8. (8 points) (Version #) Use an appropriate local linear approximation to estimate
9 Final Exam MATH 11 Fall (8 points) Determine the locations of all relative maxima or minima, if any, of f ( x) xcosxon the interval 0 x. 10. (8 points) Find the absolute maximum and minimum values of f ( x) x lnxon the interval 1,5 5. Hint: ln
10 Final Exam MATH 11 Fall (7 points) (Version #1) The graph of the derivative f '( x ) is given below. Use this graph to find all critical points of f ( x) and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. y y = f ' (x) x 1 7
11 Final Exam MATH 11 Fall (7 points) (Version #) The graph of the derivative f '( x ) is given below. Use this graph to find all critical points of f ( x) and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. y 1 y = f ' (x) x 17
12 Final Exam MATH 11 Fall (10 points) Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches. What is the maximum volume? Hint: Use similar triangles. Recall that the volume V of a right circular cylinder with radius r and height h is V r h. 8
13 Final Exam MATH 11 Fall (10 points) Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches. What is the maximum volume? Hint: Use similar triangles. Recall that the volume V of a right circular cylinder with radius r and height h is V r h. (Alternate Solution) 8
14 Final Exam MATH 11 Fall (10 points) On the axes provided on the next page, sketch the graph of the given function f and identify the locations of all critical points and inflection points. Label all intercepts and asymptotes, if any. The first and second derivatives are given to you. Hint: f ( 0.5).6 1 x 1 4 x 1 f ( x) x x f '( x) f ''( x) 5 x x 9
15 Final Exam MATH 11 Fall (ADDITIONAL SPACE FOR PROBLEM 1) Sketch the graph of the given function f and identify the locations of all critical points and inflection points. Label all intercepts and asymptotes, if any. The first and second derivatives are given to you. Hint: f ( 0.5).6 1 x 1 4 x 1 f ( x) x x f '( x) f ''( x) 5 x x 15 y x 5 10
16 Final Exam MATH 11 Fall THIS PAGE LEFT BLANK (CAN BE USED FOR EXTRA SPACE FOR PROBLEMS) 11
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