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Math 200Exam 1
May 1st, 2019
Name:
Section:
? For free response questions, you must show all work. Answers without proper justifica-tion will not receive full credit. Partial credit will be awarded for significant progress towardsthe correct answer. Cross off any work that you do not want graded.
? This is a closed-book exam. You may not use any books or notes on this exam.
? You have 50 minutes to complete this exam. When time is called, stop writingimmediately and turn in your exam at the front of the room - be sure to place your exam inthe appropriate folder.
? You may not use any electronic devices including (but not limited to) calculators,cell phones, or tablets. Using such a device will be considered a violation of the university’sacademic integrity policy and, at the very least, will result in a grade of 0 for this exam.
Page: 2 3 4 5 6 7 8 9 Total
Points: 15 15 15 15 10 15 5 10 100
Score:
Math 200 Exam 1 - Page 2 of 9 5/1/2019
1. (15 points) Find an equation for the plane containing the point A(1, 2, 3) and the line L, givenbelow.
L :
x = 3− t
y = 2 + 2t
z = t
Math 200 Exam 1 - Page 3 of 9 5/1/2019
2. (15 points) Compute all of the second-order partial derivatives of f .
f(x, y) = 3x2y4 − xey
Math 200 Exam 1 - Page 4 of 9 5/1/2019
3. Consider the vector-valued function ~r(t) = 〈3t, ln t, t2〉.(a) (8 points) Find a set of parametric equations for the line tangent to ~r at the point
A(3, 0, 1). We’ll call this line L.
(b) (7 points) Compute the angle between the line L and the plane P : x + z = 3. You mayleave your answer in terms of an inverse trig function.
Math 200 Exam 1 - Page 5 of 9 5/1/2019
4. (15 points) Determine whether the following lines are parallel, intersecting, or skew. If the linesintersect, determine the point of intersection.
L1 :
x = 2− t
y = 3t
z = 1 + 2t
L2 :
x = 4− t
y = 2 + t
z = 1 + t
Math 200 Exam 1 - Page 6 of 9 5/1/2019
5. (10 points) Match the equation with the surface. Write the letter on the line provided next toeach equation.
z = x2 + y2
x2 + y2 − z2 = −1
x2 + y2 − z2 = 1
x2 + y2 − z2 = 0
z = y2 − x2
(A) (B)
(C) (D)
(E) (F)
Math 200 Exam 1 - Page 7 of 9 5/1/2019
6. (5 points) Which of the following is an equation of the sphere that is centered at P (1, 2, 3) andis tangent to the xz-plane?
(a) (x− 1)2 + (y − 2)2 + (z − 3)2 = 1
(b) (x− 1)2 + (y − 2)2 + (z − 3)2 = 4
(c) (x− 1)2 + (y − 2)2 + (z − 3)2 = 9
(d) (x + 1)2 + (y + 2)2 + (z + 3)2 = 4
(e) (x + 1)2 + (y + 2)2 + (z + 3)2 = 9
7. (5 points) Consider the planes P1 and P2, shown below.{P1 : x + 2y + 3z = 7
P2 : 2x− 4y + kz = 11
For which value of k are the planes P1 and P2 perpendicular?
(a) k = −2
(b) k = −1
(c) k = 0
(d) k = 1
(e) k = 2
8. (5 points) Consider the vectors ~v = 〈1,−2,−1〉 and ~b = 〈−2, 1, 2〉. Which of the following isthe projection of ~v onto ~b?
(a)
⟨4
3,−2
3,−4
3
⟩
(b)
⟨−2
3,4
3,2
3
⟩(c) 〈4,−2,−4〉
(d) 〈−2, 4, 2〉
(e) 〈2,−4,−2〉
Math 200 Exam 1 - Page 8 of 9 5/1/2019
9. (5 points) Which of the following is the domain of the function f(x, y) = ln(x2 +y2−4)? Notethat for (A), (C), and (E) boundary is dotted.
(A) (B)
(C) (D)
(E)
Math 200 Exam 1 - Page 9 of 9 5/1/2019
10. (5 points) Use the following graph to answer the next two questions.
Which of the following is−−→AB +
−−→BC +
−−→CD?
(a) 〈4, 0〉
(b) 〈0, 4〉
(c) 〈6, 1〉
(d) 〈1, 6〉
(e) 〈−1, 6〉
11. (5 points) Which of the following is the vector−−→BC −
−−→CD?
(a) 〈0, 2〉
(b) 〈−1,−1〉
(c) 〈1, 1〉
(d) 〈2, 0〉
(e) 〈1,−1〉
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