Test 1, Page 1
Name:
Math 2153
Jeff Mermin’s section
Exam 1
On the essay questions (# 3–7) write legibly in complete sentences, in such a
way that I can easily tell what you are doing and why.
You may or may not find the following tables useful.
Test 1, Page 2
Name:
1. (30 points)Indicate whether the following statements are true or false.
(“True” means “Always true”, “false” means “sometimes false”.) No justification is necessary on this problem. Write the entire word “True”
or “False”. Illegible or abbreviated answers will receive no credit. In
these problems, θ is an acute angle, and f and g are continuous functions
such that f (x) ≥ 0 and g(x) ≥ 0 for all real x. (In particular, no integral
is improper except at ∞.)
Z x=∞
(a) If lim f (x) = 1, then
f (x)dx diverges.
x→∞
x=0
Z
x=∞
(b) If 0 ≤ f (x) ≤ g(x) for all x and
Z
g(x)dx
x=0
diverges.
x=∞
f (x)dx diverges, then
x=0
(c) If x = sec θ, then csc θ = x2 − 1.
Z
(d)
dx
= tan−1 (x + 2) + C.
x2 + 2x + 3
Z
(e) If f (x) > 0 for all x and lim f (x) = 0, then
x→∞
verges.
Z
(g) If x = cot θ, then csc θ =
Z
x=∞
g(x)dx diverges, then
x=0
r
f (x) dx conx=0
x=∞
(f ) If 0 < f (x) ≤ g(x) for all x and
diverges.
x=∞
x2 + 1
.
x2 − 1
f (x)dx
x=0
Test 1, Page 3
Name:
p(x)
A
B
Cx + D
= q(x) +
+
+ 2
x4 − 1
x−1 x+1
x +1
for some polynomial q and numbers A, B, C, and D.
(h) If p is a polynomial, then
Z
(i)
sec x dx = ln | tan x| + C.
Z x=∞
Z x=∞
Z x=∞
(j) If
f (x)dx and
g(x)dx converge, then
f (x)g(x) dx
x=0
x=0
x=0
converges.
Z
x=∞
xdx
con2 + 2x − 3)3
(x
x=10
verges or diverges. Justify your answer. (You don’t have to evaluate it if it
converges, but trying to evaluate it can be one way to determine whether
it converges or not.)
2. (20 points) Determine whether the integral
Test 1, Page 4
Name:
3. (20 points) State the first step you would take in computing each of the
following integrals. Our options are:
• Guess-and-fix (state the initial guess).
• Algebra (state the rewritten integral or integrals).
• u-substitution (state u and du).
• Integration by parts (state u and dv).
• Use a reduction formula (write out the entire formula, with appropriate substitutions, if any.)
• Trigonometric substitution (state x, dx, and an appropriate trigonometric identity).
• Use partial fractions (state the form of the decomposition).
• Look it up in a table (state the entry number or the integral that you
expect to find in the table, and any constants in that entry).
If instead you don’t think we can evaluate the integral with our current
techniques, say that instead.
Z p
(a)
4x2 − 1dx.
Z
(b)
Z
(c)
Z
(d)
tan5 x sec x dx.
ln(x4 − 1)dx.
x cos(9x2 )dx.
Test 1, Page 5
Name:
4. (40 points) Choose two of the three integrals below, and either evaluate
them or transform them into an equivalent but substantially easier integration problem, and explain why the new problem is easier. Make
it very clear which integrals you are choosing, for example by circling the
letters of the two chosen integrals or X-ing out the space underneath the
third integral. I will grade only two.
Z
(a)
9x(ln x)2 dx
Z
(b)
tan3 (ln x)
dx
2x
Test 1, Page 6
Z
(c)
Name:
x3 dx
x2 − 1
5. (40 points) Choose two of the three questions below, and express the solution as a definite integral. Do not attempt to evaluate the integral.
Provide some justification for your solution.
Make it very clear which questions you are choosing, for example by circling the letters of the two chosen questions or X-ing out the space underneath the third. I will grade only two.
(a) The region between the curves y = x2 and x = y 2 is rotated around
the line x = 2. Find the volume of the region it sweeps out.
Test 1, Page 7
Name:
2
(b) Find the length of the curve y = ex − 1 between the points (1, e − 1)
and (3, e9 − 1).
(c) A 75-kilogram bungee jumper leaps off a bridge, wearing a 7-kilogram
bungee cord. At the bottom of her jump, the cord is 30 meters
long. Find the work done by gravity in moving the jumper from the
bridge to the bottom of the jump. (Use the physics formulas Work =
(Force)(distance) and Force = (mass)(acceleration) and gravitational
acceleration g = 9.8 sm2 . Assume the bungee cord stretches to a uniform density, and ignore other complicating factors.)
Test 1, Page 8
Name:
6. (Extra Credit: 10 points) Write down a well-formed indefinite integral
that I can’t evaluate.
Z
7. (Extra Credit: 10 points) Evaluate
ex cos x dx by hand, without
using a table. (Hint: integrate by parts twice.)