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Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
Exam 2
MATH 131
Version A
Fall ’08
Name (printed):
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this
academic work.
Name (signature):
Section:
Seat #:
Instructions:
• You must clear your calculator: MEM (2nd +), Reset (7), cursor right to ALL,
All Memory (1), Reset (2).
• There are 16 questions and 8 pages to this exam including the cover sheet. The
multiple choice questions are worth 5 points each, and the point values for the
problems in the work out section are as indicated.
• In order to receive full credit on the work out problems, you must show appropriate, legible work.
• Round to 4 decimal places, unless otherwise stated.
• You must box or circle your final answer in the work out section.
• If you need more room to work a problem, I can provide scratch papers. Do NOT
use your own papers.
• Disputes about grades on this exam must be handled the day the exam is handed
back and must be discussed before you leave the room. If the exam leaves the
room, it will not be re-evaluated (except for possible adding mistakes).
GOOD LUCK!
Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
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Multiple Choice: Circle the letter of the correct answer.
d 99 x
1. Find the value of 99 (e ), when x = 0.
dx
a. 1
b. 2
c. 99x ex
d. 0
e. None of these.
2. Consider the equation of motion s = f (t) = 13 t 3 − 4t 2 + 12t, where t is measured in seconds and s in
meters. When is the particle at rest?
a. t = 2 and t = 6
b. t = 4
c. t = 0 and t = 4
d. t = −2 and t = −6
e. None of these
3. The position of a particle is given by the equation x = f (t) = t 4 − 2t 3 + 1 , where t is measured in
seconds (s) and x in meters (m). The acceleration of the particle at t = 2 is
a. 24 m/s
b. 24 m/s2
c. −24 m/s2
d. 48 m/s
e. None of these
Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
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4. Suppose the position of a particle is given by the equation s = 3t + 12 θt 2, where t is measured in
seconds and s in meters. Assume θ is a constant.
Then θ is:
a. velocity of the particle
b. speed of the particle
c. acceleration of the particle
d. position of the particle
e. None of these
5. Use the graph of f (x) below to answer the following question:
Which of the following statements are true? Circle ALL the true statements.
a. f (x) is both continuous and differentiable at x = 4.
b. f (x) is continuous at x = 4 but is NOT differentiable at x = 4.
c. f (x) is differentiable at x = 4 but is NOT continuous at x = 4.
d. f (x) is both continuous and differentiable at x = 0.
e. All of the above statements are false.
Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
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For the following questions, you must show appropriate, legible work to receive full credit.
6. (7 points) If f (x) = x2 + 3 , find f ′ (x) using the definition of the derivative. (Do NOT use any shortcut.
Compute f ′ (x) using the definition of the derivative.)
7. (7 points) Use calculus to determine on what interval(s) the function f (x) = 1 + ex − 4x is increasing.
Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
8. (7 points) Sketch the graph of a function that satisfies all of the given conditions:
f ′ (1) = f ′ (5) = 0
f ′ (x) > 0 if x < 1
f ′ (x) < 0 if 1 < x < 5 or if x > 5
f ′′ (x) > 0 if 3 < x < 5
f ′′ (x) < 0 if x < 3 or x > 5
9. (7 points) Find the point(s) on the curve y = 2x3 − 6x + 100 where the tangent line is horizontal.
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Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
10. (7 points) Suppose a bacteria population is given by P(t) = 30(2t ) bacteria after t hours. Find the
rate of increase of the bacteria population after 5 hours and interpret your answer.
√
11. (7 points) Find the equation of the tangent line to the curve y = cot x at the point (π/6, 3).
[Given sin(π/6) = 21 , cos(π/6) =
√
3
2 ,
tan(π/6) =
√1
3
and csc x =
1
sin x ,
sec x =
12. (5 points) The following table gives the values of some function f (t) .
1900 1910 1920 1930 1940
t
f(t) 48.3 51.1 55.2 57.4 62.5
Estimate the value of E ′ (1910).
1
cos x
, cot x =
1
tan x .]
Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
13. (7 points) Find the derivative of the following function. Do NOT simplify your answer.
f (x) = (3x2 − 2x + 7)( x45 + 4x − sec x)
14. (7 points) Use the graph of f (x) below to sketch the graph of its derivative f ′ (x).
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Math 131, Exam 2- copyright 2008 - Indranil Sen Gupta
15. (7 points) Find the derivative of the following function. Do NOT simplify your answer.
f (θ) =
99sin θ
ln(1 + θ)
16. (7 √
points) Find the linear approximation of the function f (x) =
imate 3 1.9.
√
3
2 − x at a = 0 and use it to approx-