FINAL EXAMINATION
MATH1014 (CALCULUS), SPRING SESSION, 2022
STUDENT NAME: UNI ID:
1. INSTRUCTIONS
• For full credit, you must show your work unless otherwise instructed.
• There are a total of 50 points available on this exam. The value of each question part is indicated.
• You may not use any external resources to help solve this exam (including electronic devices, books,
websites, people, etc.) except for the reference sheet provided at the end of this document.
• You have 90 minutes to write this examination.
• If participating remotely, you have 15 minutes after the writing time ends to scan and upload your
submission to Wattle. It is your responsibility to ensure that all pages have been uploaded, and to
match page and problem numbers.
Questions on next page.
Date: 6 January, 2022.
1
(1) (6 points) Give examples of the following. You do not have to justify your answers.
(a) A series
∞∑
n=0
an that does not converge, such that
∞∑
n=0
a2n converges.
(b) A series that is conditionally convergent.
(c) An improper integral that does not converge.
2
(2) (a) (7 points) Find the radius of convergence of the following power series:
∞∑
n=0
nxn
3n(n+ 1)
.
Also check convergence at endpoints. Show your work.
3
(b) (5 points) Find the power series representation of the function
1
(1− x)(1− 2x) .
Show your work.
4
(3) (7 points) Suppose that
u= x(y + 2) + x2, x = pq2, y = (q+ 1)(ln p).
Use the chain rule to find ∂ u/∂ p and ∂ u/∂ q at the point (p,q) = (e,−1).
5
(4) (10 points) Let
h(x , y) = x2 + 3y + 4y2.
Using the Lagrange Multipliers method, find the maximum and minimum values of h on the unit
circle. Show your work.
6
(5) (5 points) Evaluate the following double integral:∫∫
D
y
x5 + 1
dA where D = {(x , y) | 0≤ x ≤ 1, 0≤ y ≤ x2}.
7
(6) (10 points) Set up an iterated integral to find the volume of the solid under the surface z = 6− 5x2
and above the region enclosed by y = 2x , y = 2, x = 0 in the x y plane. Sketch the region and show
all steps of the setup, but do not evaluate the integral.
8