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〉