Math 152 Spring 2021
Final Exam Review Questions
1. The final exam may test on all material covered from the beginning of the semester up to
and including section 11.11.
We have covered quite a bit of material this term. This can be a little overwhelming so the
following list is intended to give you an idea of some of the things that I expect you to know.
This list is by no means exhaustive, but it does highlight some commons questions I have
been asked by a few students over the term.
You are expected to know:
• the antiderivatives in the table of section 7.5 (page 503 of text) - some of these can be
derived from others in the table, and some can be derived using techniques we learned
this semester. It is up to you to determine how many you should commit to memory
(probably 1 -10 should be at your fingertips ready to be used in a pinch).
• all the various techniques for integration: substitution, by-parts, trigonometric substitution (this one will not be tested), partial fractions (chapter 7)
• the trigonometric identities:
cos2 θ + sin2 θ = 1
sin (2θ) = 2 sin θ cos θ
1 + tan2 θ = sec2 θ
cos (2θ) = cos2 θ − sin2 θ = 2 cos2 θ − 1 = 1 − 2 sin2 θ
• the Midpoint, Trapezoid and Simpson’s Rule and how to apply them. You will not be
required to memorize the error bounds. But you may need to know how to use them, in
which case they will be provided in the question.
• the formulas for arc length (section 8.1), surface area (section 8.2)
• area and arc length for curves given by parametric equations and area in polar coordinates (sections 10.2, 10.4)
• area of a surface of revolution for a curve given by parametric equations (sections 10.2)
• Series Tests: test for divergence, integral test, comparison test, limit comparison test,
alternating series test, ratio test, root test (sections 11.2 - 11.7) (See chart in notes for
section 11.7 for the ”big picture”.)
• techniques for approximating series - though I won’t expect you to do anything that
would require the use of a calculator
• the definition of Taylor Series and Maclaurin Series and how to compute these series for
a given function (section 11.10 )
Make sure you review ALL the questions from the homework assignments and midterms. It
is expected that you will know how to do all of these at the time of the final exam.
The following questions can be thought of as a sample exam. The best way to use this resource
is to sit down in a distraction free place, put your notes and textbook away, and then pretend
you are actually writing the exam. Once finished, you should have an idea of what you still
need to practice.
Solutions for the following questions will not be posted, though answers are included on the
last page. If you want to determine whether you have done a question correctly then you can
visit the TA’s or me in the workshop.
1
2. Compute the following integrals.
(a)
(b)
(c)
(d)
!
!
!
!
6
x2
2
x+1
dx
+ 2x − 3
ex
dx
e2x + 2ex + 2
e
√
3
t
dt
x ln x
√
dx
x2 − 1
3. Find the solution to the differential equation that satisfies the initial condition:
dy
y cos x
=
,
dx
1 + y2
y(0) = 1.
You may leave the solution in implicit form.
4. Find the length of the curve
y=
!
1
x"
t2 − 1 dt, 1 ≤ x ≤ 4.
5. (a) The curve in the figure is called an Archimedean spiral and it has parametrization
x = t cos t, y = t sin t
0 ≤ t ≤ 2π.
Compute the length of the curve.
! "
$
"
1# " 2
t t + 1 + ln(t + t2 + 1) .
2
(b) Set up, but do not evaluate, the integral for the area
of the region bounded by the Archimedean spiral and
the x-axis as shaded in the figure.
Note: You may use
t2 + 1 dt =
(c) Set up, but do not evaluate, the integral for the area of the surface of revolution obtained
by rotating the part of the spiral in the second quadrant about the x-axis.
2
6. (a) Let R denote the region bounded by the curve
y = x2 (x − 2)
(0 ≤ x ≤ 2)
and the x-axis. Using the method of cylindrical shells
compute an exact value for the volume of the solid of
revolution obtained by rotating R about the y-axis.
Draw a typical cylindrical shell in the diagram.
(b) Formulate, but do not evaluate, an integral representing the volume of the solid of
revolution obtained by rotating R about x = 3. Sketch a diagram and indicate which
method you are using: Washer Method or Shell Method.
7. For each series determine whether if is convergent or divergent. In each case, state the test(s)
you are using, justify the steps in using the test, and clearly indicate whether the series is
convergent or divergent.
(a)
(b)
(c)
∞
%
n=1
∞
%
n=1
∞
%
√
n2 − 1
n3 + 1
n!
+ 2)!
2n (n
(−1)n
n=2
ln n
n
8. For each of the following power series compute the radius R of convergence and the interval
I of convergence. Justify your answer.
(a)
(b)
∞
%
n=1
∞
%
n=1
(−1)n
xn
n2 7n
2n (x − 2)n
(n + 2)!
9. Prove that the series
∞
%
n=2
is convergent for any p > 1.
1
n(ln n)p
10. (a) Define the Taylor Series of a function f at a.
(b) Compute the Taylor series for the function f (x) = cos (3x) at a = π/2.
! b
11. Suppose you want to approximate the integral
f (x) dx where f is the function given
0
in the figure. You decide to use the Riemann Sum with 4 subdivisions (i.e. n = 4), but
are debating whether to use a right-hand approximation R4 or the Trapezoid rule T4 . Which
3
approximation will be more accurate? Give an explanation supporting your answer and sketch
the approximating areas (rectangles or trapezoids) in the diagram.
4
Answers:
Only answers are provided here. You are expected to provide fully worked out solutions. If you
need help with solving any of these problems please visit the Calculus Workshop.
√
√
x + 1) + C
3 t (t2/3 − 2t1/3 + 2) + C)
2. (a)
ln
3
(b)
arctan
(e
(c)
3e
(d)
x2 − 1 ln x −
√
−1
2
x − 1 + sec x + C
3. ln |y| + 12 y 2 = sin x +
4.
1
2
15
2
5. (a) π
6. (a)
√
4π 2
16π
5
+
1+ 12
(b) π
7. (a) converges
!
ln (2π +
2
0
√
4π 2
+ 1)
(b)
!
2π
π
−t sin t(cos t−t sin t) dt
(c)
!
π
2πt sin t
π/2
6x(2 − x) + x2 (2 − x)2 dx
(b) converges
8. (a) R = 7, I = [−7, 7]
(c) converges
(b) R = ∞, I = (−∞, ∞)
9. (a) See definition in section 11.10 of textbook.
(b)
∞
%
(−1)n 32n+1 (x − π2 )2n+1
(2n + 1)!
n=0
5
"
1 + t2 dt