MA Calculus III Exam 3 : Part I 25 November 2013

Transcription

1 MA Calculus III Exam 3 : Part I 25 November 2013 Instructions: You have as long as you need to work on the first portion of this exam. When you finish, turn it in and only then you are allowed to use your calculator for the remaining portion of this exam. You may not use any outside assistance on this exam. You may not use books, notebooks, other people s exams, cell phones, mp3 players, or any other materials to cheat on this exam. You may not use graphing calculator on this portion of the exam. If you are caught cheating on either portion of the exam, you will be given a 0 for both portions. You must write clearly, give exact answers and fully reduce fractions to receive full credit. Unreadable, approximate and unreduced answers will receive only partial credit. You must show all your work to receive full credit unless otherwise stated. Name: Score: /100 Points 1

2 SCRATCH WORK 2

3 1. (10 points ) Evaluate the following limit if it exists. If it does not exists, explain why not. x 4 lim (x,y) (0,0) x 4 + y 4 3

4 2. Consider the function f(x, y) = x 2 + 4y 2. (a) (5 points) What is the domain of f(x, y)? (b) (5 points) What is the range of f(x, y)? (c) (10 points) Draw a contour diagram for f(x, y) showing at least four contour lines for integer values of z. 4

5 3. (3 points each) True or False: Circle one. You do not have to provide any explanation. (a) T F The projection of the curve defined by r(t) = cos ti+sin tj+2tk onto the xz-plane is a circle. (b) T F If r is the position vector of a particle moving along a smooth curve in space, then v(t) = dr is the particle s velocity vector. dt (c) T F If T is the unit tangent vector of a smooth curve, then the curvature of the curve is given by: κ = dt. dt (d) T F If r(t) defines a smooth curve that lies on the surface of a cylinder, then the principle unit normal for the curve always points to the outside of the cylinder. (e) T F Thanksgiving is Dr. Kahn s favorite holiday. (Hint: He loves turkey, potatoes, gravy, and all holidays in November.) 5

6 MA Calculus III Exam 3 : Part II Instructions: You have the remainder of the period to finish portion II of this exam. You may use your graphing calculator on this portion of the exam after you have turned in Portion I. You must show your work setting up each problem before and after using the calculator. Proper justification of all work is still required. Name: 1. (10 points) Solve the initial value problem: dr dt = 3 2 (t + 1)1/2 i + e t j + 1 k, r(0) = k. t + 1 1

7 2. Consider the function r(t) = (e t cos t)i + (e t sin t)j + 2k. (a) (10 points) Calculate the unit tangent vector T at t = π/4. (b) (10 points) Calculate the principal unit normal vector N at t = π/4. 2

8 3. (5 points each) Consider the contour diagram below that represents a function f(x, y). The contours displayed reading from the center out are for: z = 4, 3, 2, 1, 0, 1, 2 over the intervals [ 5, 5] and [ 5, 5] for x and y respectively (a) Sketch the vector of steepest incline from the point ( 1.6, 0) onto the contour diagram and label it v. (b) How many peaks from the surface of f(x, y) are represented on the contour diagram? Is there a tallest peak and if so in which square does it reside? (c) How many valleys from the surface of f(x, y) are represented on the contour diagram? 3

9 4. (10 points) Determine the arclength of the curve over the interval [1, 2] defined by: r(t) = 6t 3 i 2t 3 j 3t 3 k. 5. (5 points) Extra Credit: Find the point of maximum curvature of the planar curve y = e x. 4