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University of Connecticut
Department of Mathematics
Math 1131
Sample Exam 2
Fall 2013
Name:
Instructor Name:
Section:
TA Name:
Discussion Section:
This sample exam is just a guide to prepare for the actual exam. Questions on
the actual exam may or may not be of the same type, nature, or even points. Don’t
prepare only by taking this sample exam. You also need to review your class notes,
homework and quizzes on WebAssign, quizzes in discussion section, and worksheets.
The exam will cover from section 3.4 through section 4.7.
Read This First!
• Please read each question carefully. Other than the question of true/false items, show all
work clearly in the space provided. In order to receive full credit on a problem, solution
methods must be complete, logical and understandable.
• Answers must be clearly labeled in the spaces provided after each question. Please cross out
or fully erase any work that you do not want graded. The point value of each question is
indicated after its statement. No books or other references are permitted.
• Unless instructed otherwise, give any numerical answers in exact form, not as approximations. For example, one-third is 31 , not .33 or .33333. And one-half of π is 12 π, not 1.57 or
1.57079.
• Turn smart phones, cell phones, and other electronic devices off (not just in sleep mode) and
store them away.
• Calculators are allowed but you must show all your work in order to receive credit on the
problem.
• If you finish early then you can hand in your exam early.
Grading - For Administrative Use Only
Question:
1
2
3
4
5
6
7
8
9
10
11
Total
Points:
15
7
7
9
10
6
8
10
10
9
9
100
Score:
Math 1131
Sample Exam 2
1. If the statement is always true, circle the printed capital T. If the statement is sometimes
false, circle the printed capital F. In each case, write a careful and clear justification or a
counterexample.
(a) If x sin x has a local maximum value at x = c then tan c = −c.
T
F
[3]
T
F
[3]
T
F
[3]
T
F
[3]
T
F
[3]
Justification:
(b) If f (x) is a differentiable function then the derivative of f (ex ) is f 0 (ex )ex .
Justification:
(c) Every continuous function on (0, 1) has an absolute maximum value.
Justification:
(d) If
dy
= y then y = 0 or y = ex .
dx
Justification:
(e) The graph of y = ln(x2 ) for x > 0 is concave down.
Justification:
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Math 1131
2. (a) Compute
Sample Exam 2
d
x ln2 (x) . You do not need to simplify.
dx
(b) Find the derivative of y = 5x in two ways and check they agree:
(i) logarithmic differentiation, writing the final answer entirely in terms of x.
(ii) express 5x as a power of e and use the fact that (ex )0 = ex .
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[3]
[4]
Math 1131
Sample Exam 2
3. Use implicit differentiation to find the equation of the tangent line to the graph of y 2 = x3 +2xy
at the point (3, −3), as marked below. Write the equation in the form y = mx + b.
y
x
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[7]
Math 1131
4. (a) Find the linearization of
Sample Exam 2
x+3
at the point (0, 3).
2x + 1
(b) The side length L of a cube is measured to be 20 cm, with an error of at most .3 cm. Use
differentials to estimate the maximum percentage error in using L = 20 to compute
(i) the volume of the cube,
(ii) the surface area of the cube.
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[3]
[6]
Math 1131
Sample Exam 2
5. (a) Two boats leave a dock at the same time. One boat travels south at 30 mi/hr and the
other travels west at 40 mi/hr. After half an hour, how quickly is the distance between
the boats increasing, in mi/hr?
[5]
(b) A spy plane is flying 500 m above the ground at 450 km/hr, and its path goes directly
over an enemy tracking station that is already tracking it.
(i) How many meters does the plane cover in two seconds?
(ii) Determine how quickly the angle between the ground and the line from the tracking
station to the plane is changing, in radians per second, two seconds after the plane flies
over the tracking station.
[5]
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Math 1131
Sample Exam 2
6. A pile of the radioactive substance Unobtainium loses 6% of its mass in a year.
(a) If a sample of Unobtainium has an initial mass of 50 grams, determine a formula for U (t),
the amount of Unobtainium left in the sample after t years.
[3]
(b) Find the half-life of Unobtainium in years, accurate to 3 decimal places.
[3]
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Math 1131
Sample Exam 2
7. Use calculus to find the absolute maximum and minimum values of the following functions
on the indicated intervals. Answers can be given to three decimal places.
(a) f (x) = sin x + cos x, on [0, π]
[4]
(b) f (x) = (7x − 1)e−2x , on [0, 1]
[4]
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Math 1131
8. When f (x) =
Sample Exam 2
x
, use calculus to find
x2 + 1
(i) the critical numbers of f (x),
(ii) the open intervals where f (x) is increasing and where f (x) is decreasing,
(iii) the open intervals where the graph of y = f (x) is concave up and concave down.
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[10]
Math 1131
Sample Exam 2
9. (a) Find the points on the curve y = x2 whose distance to the point (0, 1) is minimal.
(b) A box-shaped shipping crate with a square base needs to have a volume of 80 ft3 . The
material used to make the base of the crate costs twice as much (per ft2 ) as the material
used for the sides, and the material used to make the top of the crate costs half as much
(per ft2 ) as the material used for the sides. Use calculus to find the dimensions of the
crate that minimize the total cost of the materials.
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[5]
[5]
Math 1131
Sample Exam 2
10. Here are three theorems about continuous functions. Draw a picture that illustrates each
theorem, using the notation of the theorem in your picture.
(a) Extreme Value Theorem: If f (x) is a continuous function on [a, b] then it has an
absolute maximum value and an absolute minimum value on [a, b].
[3]
(b) Rolle’s Theorem: If f (x) is a continuous function on [a, b] that is differentiable on (a, b),
and f (a) = f (b), there is a c ∈ (a, b) such that f 0 (c) = 0.
[3]
(c) Mean Value Theorem: If f (x) is a continuous function on [a, b] that is differentiable
f (b) − f (a)
on (a, b), there is a c ∈ (a, b) such that f 0 (c) =
.
b−a
Draw a picture for the case when f (a) 6= f (b).
[3]
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Math 1131
Sample Exam 2
11. Evaluate the following limits using l’Hˆopital’s Rule.
5x − 4x
(a) lim x
x→0 3 − 2x
sin2 (ax)
, where a 6= 0. (The answer will depend on a.)
x→0
x2
(b) lim
(c) lim
x→∞
1+
10 x2
x2
[3]
[3]
[3]
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