Math 2374 Final Exam Fall 2020
Thursday, December 17, 2020
Time limit: 90 minutes
Instructions – read carefully
This exam has 8 questions, worth a total of 60 points. You have 90 minutes to complete this exam.
You may either print this document and write directly onto it, or you can use your own paper and
upload your responses. Tablet/stylus is also acceptable. Your TA will announce when 90 minutes
have passed; at this point you must stop working on the exam and begin uploading your solutions.
You will then have 15 minutes to complete the upload. The upload must be completed by 1:50pm.
What you can use: Textbook, lecture notes, and any other notes/formula sheets you have prepared
yourself.
What you CANNOT use: Everything else. In particular, no calculators, no graphing calculators,
and Mathematica is not allowed. Also, you may not use any online resources including, but not
limited to, Chegg, Google, Wolfram Alpha, etc. You may not communicate (electronically or
otherwise) with any other students in the class.
Uploading: Please upload your responses to Gradescope, and be sure to upload to the assignment
labelled according to your discussion section. Make sure that each question is answered on a different
page – if a question has multiple parts (a), (b), etc. you may use the same page for all parts. Be
sure to correctly assign pages to problems in Gradescope, otherwise they will not be
graded accurately.
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Total
8 6 6 8 8 8 8 8 60 points
1. (2 points each) Choose one response for each of the four multiple choice questions below.
(a) The function f (x, y, z) = x2 + y 2 + z 2 − xz − 1 has one critical point. It would be classified
as a
(i)
(ii)
(iii)
(iv)
local maximum
local minimum
saddle
none of the above
(b) Which of the following lines is perpendicular to the tangent plane to the surface x2 y − z = 1
at the point (1, 2, 1)?
(i)
(ii)
(iii)
(iv)
`(t) = h1 + 4t, 2 − t, 1 + 4ti
`(t) = h1 + 2t, 2 − t, 1 + 4ti
`(t) = h1 + 4t, 2 + t, 1 − ti
`(t) = h1 + 2t, 2 + 2t, 1 − 4ti
(c) Consider the surface S, which is the portion of the graph z = cos(xy) + x2 in the region
x2 + y 2 ≤ 1, and ∂S is its boundary. For which of the following orientations do we have that
ZZ
Z
∇ × F · dS =
F · ds
S
∂S
(i) S is oriented with upward normal, and ∂S is oriented counterclockwise (viewed from
above).
(ii) S is oriented with downward normal, and ∂S is oriented counterclockwise (viewed from
above).
(iii) S is oriented with upward normal, and ∂S is oriented clockwise (viewed from above).
(iv) all of the above
(d) Consider the function f (x, y) = 2x + y 2 . Suppose the curve C(t) satisfies C(0) = h4, −1i
and C 0 (0) = h−2, 1i. Then D(f ◦ C)(0) is equal to
(i)
(ii)
(iii)
(iv)
-6
10
-2
none of the above
2. (6 points) Find a second order approximation for the function f (x, y) = 2ey cos x near (0, 0), and
use this to estimate the quantity 2e0.1 cos(0.1).
3. (6 points) Find the surface area of the portion of the sphere x2 + y 2 + z 2 = 4 which is inside the
cylinder x2 + y 2 = 1 in region z ≥ 0.
4. (8 points) Consider the vector field F(x, y, z) = h2x − y, sin(y 2 ) − x, z − 4z 2 ln(1 + z 2 )i.
(a) Show that F is conservative.
Z
(b) Evaluate
C
F · ds where C is the curve C(t) = h2t2 , cos(2πt), 3 sin(2πt)i where 0 ≤ t ≤ 1.
Hint: Try a different curve with the same endpoints.
5. (8 points) Find and classify any critical points of the function f (x, y) = 4x2 − 2xy + y 2 − 1
using the second derivative test where appropriate. Then determine the global maximum and
minimum of f inside the ellipse 4x2 + y 2 ≤ 4.
Z
16 Z 2
6. (8 points) Find
0
y 1/4
2
e−y/x
dxdy.
x
7. (8 points) Find the area of the region D bounded by the y-axis and the part of the curve
C(t) = h1Z− cos t, t + cos ti between t = 0 and t = 2π. Hint: try to relate this to an integral
involving
xdy.
ZZ
8. (8 points) Compute the flux
S
2
F·dS of the vector field F(x, y, z) = h−5ey z , cos(x+z), x2 +y 2 i
out of the “fishbowl” shaped surface S, which is the portion of the ellipsoid x2 + y 2 + 2z 2 = 6
lying below the plane z = 1. Hint: Use Gauss’ theorem.