MATH 2263
Name (Print):
Spring 2022
Student ID:
Midterm 1
Section Number:
Feb 15, 2022
Teaching Assistant:
Time Limit: 50 minutes
Signature:
This exam contains 7 pages (including this cover page) and 6 problems. Check to see if any pages are missing.
Enter all requested information on the top of this page, and put your initials on the top of every page, in
case the pages become separated.
Do not give numerical approximations
to quantities such as sin(5), π, or
√
quantities such as cos(π/4) = 2/2, e0 = 1, and so on.
√
2. However, you should simplify
The following rules apply:
• Show your work, in a neat and coherent way, in the space provided. All answers must be justified by
mathematical reasoning, including the evaluation of derivatives. To receive full credit on a problem,
you must show enough work so that your solution can be followed by someone without a calculator.
• Mysterious or unsupported answers will not receive full credit. Your work should be mathematically
correct and carefully and legibly written.
• A correct answer, unsupported by calculations or explanation will receive no credit; an incorrect answer
supported by substantially correct calculations and explanations will receive partial credit.
Question
Points
1
10
2
20
3
20
4
15
5
15
6
20
Total:
100
Score
MATH 2263
Midterm 1, Page 2 of 7
1. Consider vectors a = ⟨4, −3, 1⟩ and b = ⟨2, 5, 2⟩.
(a) (5 points) Find the dot product a · b.
(b) (5 points) Find the scalar projection of b onto a.
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MATH 2263
Midterm 1, Page 3 of 7
2. Consider the following three points in R3 :
P (−1, 2, 3),
Q(2, 7, 4),
Feb 15, 2022
R(5, −2, 0).
(a) (15 points) Find a scalar equation of the plane Σ passing through P, Q, R.
(b) (5 points) Find the intersection point of the plane Σ with the y-axis.
MATH 2263
Midterm 1, Page 4 of 7
3. (a) (5 points) Evaluate the following limit, or explain why it does not exist:
9x2 − xy 2
.
(x,y)→(1,2) 3x2 + 2y 2
lim
Justify how you obtained your answer.
(b) (15 points) Evaluate the following limit, or explain why it does not exist:
9x2 − xy 2
.
(x,y)→(0,0) 3x2 + 2y 2
lim
Justify how you obtained your answer.
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MATH 2263
4. (15 points) Consider the function
Midterm 1, Page 5 of 7
f (x, y) = x sin(xy) + 5x2 y
Find the second partial derivatives fxx and fxy .
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MATH 2263
Midterm 1, Page 6 of 7
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5. (15 points) Suppose that z = f (x, y) is a differentiable function of x, y satisfying fx (1, 3) = 5 and
fy (1, 3) = 4. If x and y are functions of s and t given by
x(s, t) = s2 t
Find ∂z
∂s at (s, t) = (−1, 1).
y(s, t) = −3s.
MATH 2263
Midterm 1, Page 7 of 7
6. For the following questions, consider the function
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f (x, y) = 3y 2 + cos(x) + 9y + 5.
(a) (16 points) Find the rate of change of f at the point P = (0, 0) in the direction of ⃗v = ⟨3, 2⟩
(Caution: is ⃗v a unit vector?).
(b) (2 points) In which direction does f increase fastest at P ?
(c) (2 points) Find the maximum rate of change of f at P .