Math 1131 Sample Final Exam Spring 2015 Name:

University of Connecticut
Department of Mathematics
Math 1131
Sample Final Exam
Spring 2015
Name:
Instructor Name:
Section:
TA Name:
Discussion Section:
Read This First!
• Please read each question carefully.
• The questions on the practice finals are just for your review. You should look at all the other
practice exams and midterm 1 and midterm 2 when preparing for the final exam. The final
exam is a comprehensive exam which will cover topics that are covered during the entire
course. DO NOT MAKE THE MISTAKE of assuming only those topics on the final practice
will be on the actual final exam.
Grading - For Administrative Use Only
Question:
1
2
3
4
5
6
7
8
9
10
Total
Points:
15
8
12
12
8
8
10
9
10
8
100
Score:
Math 1131
Sample Final Exam
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) ln(23x ) = 3 ln(2x ).
T
F
[3]
T
F
[3]
T
F
[3]
T
F
[3]
T
F
[3]
Justification:
(b)
d
(sin3 x) = 3 sin2 x cos x
dx
Justification:
(c) If f 0 (x) = ln x then (f (x2 ))0 = 4x ln x.
Justification:
Z
(d)
2x
dx = ln(x2 + 1) + C.
x2 + 1
Justification:
d
(e) If
dx
Z
5
x2 dx = x2 .
2
Justification:
Page 1 of 10
Math 1131
Sample Final Exam
2. Use calculus to compute the following limits. If the limit does not exist, write DNE.
4x − x4
x→2 x2 − 4
(a) lim
(b) lim
x→∞
[4]
1000 ln x
x3
[4]
Page 2 of 10
Math 1131
Sample Final Exam
3. Compute antiderivatives of the following functions using antidifferentiation rules.
1
(a) f (x) = ekx + 2 , where k is a nonzero constant.
x
[4]
√
(b) f (x) = x x2 − 5
[4]
(c) f (x) = (sin x)(cos2 x)
[4]
Page 3 of 10
Math 1131
Sample Final Exam
4. Integration.
2
Z
e3x dx
[4]
(x3 − x) dx, where b is a constant.
[4]
p
x2 − 1 dx as a definite integral in terms of u = x2 − 1, but do not evaluate
[4]
(a) Compute
0
b
Z
(b) Compute
1
Z
(c) Express
3
x
1
the integral.
Page 4 of 10
Math 1131
Sample Final Exam
5. Use calculus to find the equation of the tangent line to the curve xy + y 3 = 14 at the point
(3, 2).
y
x
Page 5 of 10
[8]
Math 1131
Sample Final Exam
6. A helicopter rises vertically so that at time t its height is h(t) = t2 + t, where t is measured in
seconds and h(t) is measured in meters. At the time when the height of the helicopter is 20,
what is its velocity (in m/sec) and acceleration (in m/sec2 )? (Hint: First you need to find the
time when the height is 20.)
Page 6 of 10
[8]
Math 1131
Sample Final Exam
7. Let f (x) = x4 − 8x2 . Use calculus to find the open intervals on which f is increasing or
decreasing, the local maximum and minimum values of f , the intervals of concavity and the
inflection points.
Page 7 of 10
[10]
Math 1131
Sample Final Exam
8. (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?
[4]
(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]
Page 8 of 10
Math 1131
Sample Final Exam
9. After cutting out square corners from a square piece of cardboard, as in the figure below, what
remains can be folded up to form a rectangular box with an open top. Find the dimensions of
the original rectangular cardboard and the side length of the cut corners that produces a box
with volume 36 in3 and requiring the least amount of initial cardboard.
Page 9 of 10
[10]
Math 1131
Sample Final Exam
R3
10. In the figure below, draw rectangles in the Riemann sum approximation to 1 f (x) dx using 4
rectangles with right endpoints. Label numerically where the sides of the rectangles meet the
x-axis.
y = f (x)
y
x
1
Page 10 of 10
3
[8]