lieberman exam 1

7/30/2019 Lieberman Exam 1

1/8

Name:

Recitation:

MATH 114 - Spring 2013Midterm Exam I

You may use both sides of a 8.5 11 sheet of paper for notes while you take this exam. No calcu-lators, no phones, no course notes, no books, no help from your neighbors. Show all work, evenon multiple choice or short answer questionswe will be grading more on the basis of work

shown than on the end result. Remember to put your name and recitation number at the top ofthis page. Good luck.

Problem Score (out of)

1 (10)

2 (10)

3 (10)

4 (10)

5 (10)

6 (9)

7 (11)

Total (70)

By my signature below, I affirm that I have complied with the Penn Code of AcademicIntegrity in completing this exam.


2/8

1. (10 points) Find the point on the plane given by x + 2y 3z = 5 that is closest to the point(1, 1,2).(A) (1, 1,

2)

(B) (1/3, 1/3,4/3)(C) (2/3, 2/3,1)(D) (2, 3,1)(E) (2/7, 3/14,10/7)(F) (5/7, 3/7,8/7)


3/8

2. (10 points) Compute the curvature of the plane curve parametrized by

r(t) =

2cos t 2, 1 +

2sin t

at the point (1, 2).(A) 0

(B)

2

(C) 1/

2

(D) 2

(E) 1/2

(F) Undefined


4/8

3. (10 points) Compute the distance from the point (1, 0, 0) to the line tangent to the curveparametrized by

r(t) = 2 + et cos(t), t2 + t 1, ln(1 + t)at the point (3,1, 0).(A) 0

(B)

3

(C)

14

(D)

14/3

(E) 1/

3

(F) 3


5/8

4. (10 points) For times t > 0, a moving object is subject to acceleration a(t) = 6t,1/t2, 2, andat time t = 1 it has velocity 1, 0, 2 and position (0, 0, 1). At time t = 2, its position is:(A) (1, 0, 3)

(B) (12,1/4, 2)(C) (5, ln 2 1, 4)(D) (4, ln 2 2, 3)(E) (15, 3 + ln 2, 4(F) (0, 0, 0)


6/8

5. (10 points) Let f(x, y) = (x 1)2 + (y + 2)2, and let C be the curve parametrized by r(t) =2 + t,3 t. Which of the following are true?I. Each level curve of f has constant curvature.

II. C is perpendicular to the level curves of f.

III. The arc length parametrization of C is r(s) = 2 + 2s,3 2s.(A) I only (B) II only (C) III only (D) I and II (E) II and III (F) I, II, and III


7/8

6. Let C be the curve parametrized by r(t) = 43

t3/2, t + 1, t2 with 0 t 5.(i) (5 points) What is the length of the portion of the curve between (4/3, 2, 1) and (32/3, 5, 16)?

(ii) (4 points) Find an expression for the angle between C and the plane x + y + z = 1 at their pointof intersection: (0, 1, 0).


8/8

7. Suppose that the temperature at each point in the first octant (x, y, and z all positive) is givenby T(x,y,z) = x2 xy + y.(i) (5 points) If we move along the path described by r(t) =

t, et

, what is the rate of change of

T in our direction of motion at time t = 1?

(ii) (3 points) If we found ourselves at the point (1 , 3), in what direction should we move to producethe most rapid decrease in temperature?

(iii) (3 points) What is the derivative of T in the direction of 2, 1 at the point (1, 3)?

lieberman exam 1

Documents