MATH 121 FINAL EXAM FALL 2010-2011
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 PAGE 2 PAGE 3 PAGE 4 PAGE 5
24 points 12 points 12 points 18 points 16 points
Score
Points PAGE 6 PAGE 7 PAGE 8 PAGES 9-10
16 points 7 points 10 points 10 points
Score
Raw Score (out of 125): ____________
Final Score (100 * Raw Score / 125 ): _____________
Final Exam MATH 121 Fall 2010‐2011
dy
1. (6 points each) Compute .
dx
2
a. y 5 x3 2
x x ex
x
b. y x 5 tan 1 x
c. y 2 cos 4 3 x 2
d. y x 2 1
x
1
Final Exam MATH 121 Fall 2010‐2011
x
2. (6 points) Find all values of x at which the tangent line to the curve y is horizontal.
x 9
2
dy x 2 d2y
3. (6 points) Suppose that y is an implicit function of x and that . Express in terms
dx y 2 dx 2
of x and y.
2
Final Exam MATH 121 Fall 2010‐2011
4. (3 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 s (t ) t 3 feet.
2
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 2 5, x2
f ( x) m( x 1) k , 1 x 2
2 x3 x 7, x 1
3
Final Exam MATH 121 Fall 2010‐2011
4. (3 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 s (t ) t 3 feet.
2
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.
3x3 x 5, x 1
f ( x) m k ( x 3), 3 x 1
x 2 7, x 3
3
Final Exam MATH 121 Fall 2010‐2011
6. (6 points each) Compute the limits.
x2 9
a. lim
x 3 x 3
4x2 2
b. lim
x x3
1
1 x
c. lim x
x 1
4
Final Exam MATH 121 Fall 2010‐2011
7. (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1 cm 2 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 2 .
8. (8 points) (Version #1) Use an appropriate local linear approximation to estimate 2.02 .
3
5
Final Exam MATH 121 Fall 2010‐2011
7. (8 points) A spherical snowball melts so that its surface area decreases at a rate of 1 cm 2 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 2 .
8. (8 points) (Version #2) Use an appropriate local linear approximation to estimate 4.04 .
2
5
Final Exam MATH 121 Fall 2010‐2011
9. (8 points) Determine the locations of all relative maxima or minima, if any, of
f ( x) x 2 cos x on the interval 0 x 2 .
10. (8 points) Find the absolute maximum and minimum values of f ( x) x ln x on the interval
1
5 ,5 . Hint: ln 5 1.6 .
6
Final Exam MATH 121 Fall 2010‐2011
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.
y
3
y = f ' (x)
-1 1 2 3 4 5 6 7
-1
7
Final Exam MATH 121 Fall 2010‐2011
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.
y
3
y = f ' (x)
-2 -1 1 2 3 4 5 6
-1
-2
7
Final Exam MATH 121 Fall 2010‐2011
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 2 h .
8
Final Exam MATH 121 Fall 2010‐2011
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 2 h .
(Alternate Solution)
8
Final Exam MATH 121 Fall 2010‐2011
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: f (0.5) 3.6
1
2 2 x 1 4 x 1
f ( x) 3 x 3 2 x f '( x) 2
f ''( x) 5
x 3 3x 3
9
Final Exam MATH 121 Fall 2010‐2011
(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: f (0.5) 3.6
1
2 2 x 1 4 x 1
f ( x) 3 x 3 2 x f '( x) 2
f ''( x) 5
x 3 3x 3
y
15
10
-3 -2 -1 1 2
-5
10
Final Exam MATH 121 Fall 2010‐2011
THIS PAGE LEFT BLANK
(CAN BE USED FOR EXTRA SPACE FOR PROBLEMS)
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