MATH 121 FINAL EXAM FALL December 6, 2010

December 6, 2010

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 24 points PAGE 2 12 points PAGE 3 12 points PAGE 4 18 points PAGE 5 16 points Score Points PAGE 6 16 points PAGE 7 7 points PAGE 8 10 points PAGES 9-10 10 points Score

Raw Score (out of 125): ____________

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1. (6 points each) Compute dy

dx. a. y 5x3 2₂ x x ex x     b. yx5tan1x c. y2 cos4

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2. (6 points) Find all values of x at which the tangent line to the curve ₂ 9 x y x   is horizontal.

3. (6 points) Suppose that y is an implicit function of x and that

2 2 dy x dx  y . Express 2 2 d y dx in terms of x and y.

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4. (3 points each) During the first 10 seconds of a rocket flight, the rocket is propelled straight up so that in t seconds it reaches a height of ( ) 1 3

2

s t  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.

2 3 5, 2 ( ) ( 1) , 1 2 2 7, 1 x x f x m x k x x x x     _       _{   } 

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4. (3 points each) During the first 10 seconds of a rocket flight, the rocket is propelled straight up so that in t seconds it reaches a height of ( ) 1 3

2

s t  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 #2) Find values of the constants k and m that will make the following

function continuous everywhere.

3 2 3 5, 1 ( ) ( 3), 3 1 7, 3 x x x f x m k x x x x      _          

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6. (6 points each) Compute the limits.

a. 2 3 9 lim 3 x x x    b. 2 4 2 lim 3 x x x    c. 1 1 1 lim x x x     _    

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7. (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1cm min2 . 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r2.

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7. (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1cm min2 . 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r2.

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9. (8 points) Determine the locations of all relative maxima or minima, if any, of

( ) 2 cos

f x  x xon the interval 0 x 2.

10. (8 points) Find the absolute maximum and minimum values of ( )f x  x lnxon the interval 1 ,5 5      . Hint: ln 5 1.6 .

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11. (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.

-1 1 2 3 4 5 6 7 -1 1 2 3 x y y = f ' (x)      

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11. (7 points) (Version #2) 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.

-2 -1 1 2 3 4 5 6 -2 -1 1 2 3 x y y = f ' (x)      

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12. (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.

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12. (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 h2 . (Alternate Solution)

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13. (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: ( 0.5)f   3.6

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(ADDITIONAL SPACE FOR PROBLEM 13)

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: ( 0.5)f   3.6

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