Math 116 First Midterm October 7, 2014

Transcription

1 Math 116 First Midterm October 7, 2014 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 10 pages including this cover AND IS DOUBLE SIDED. There are 9 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions. 5. Show an appropriate amount of work (including appropriate explanation). Include units in your answer where that is appropriate. Time is of course a consideration, but do not provide no work except when specified. 6. You may use no aids (e.g., calculators or notecards) on this exam. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph that you use. 8. Turn off all cell phones and pagers, and remove all headphones and hats. 9. Remember that this is a chance to show what you ve learned, and that the questions are just prompts. Problem Points Score Total 100

2 Put your name on your exam and turn it in with your exam book. Write all of your answers in the exam book. Label problems clearly and circle final answers. Correct answers accompanied by incorrect or incomplete work will not receive full credit. Good Luck! Math 116 / Exam 1 (October 7, 2014) page 2 points)for Foreach eachof of the the following following equations, equations, identify name of theand surface, it with [18(15 points] match itthe with a sketch withmatch a set of level a sketch, and match it with set of levelon curves. (It is unused not necessary to showsketch work the for this question. curves (explain your athoughts). the three level curves, gradient of thebut if you get the question wrong, some work might be worth partial credit.) function at the point (1, 1). Explain how you got this. (a) 2x2 3y 2 (b) 4y 2 z2 = 0 4x2 + (c) 3x2 + y 2 + 4z 2 = 1 (i) (ii) (iii) (iv) (v) (vi) z=1 (U) z=-1 z=-1 z=1 z=1 z=1, z=-1 (V) (W) z=-1 z=.25, z=-.25 z=1 z=1 z=-1 (X) (Y) (Z) z=.25, z=-.25

3 Math 116 / Exam 1 (October 7, 2014) page 3 This page left blank for explanations.

4 Math 116 / Exam 1 (October 7, 2014) page 4 2. [18 points] Let f(x, y) = x 2 y 2 x. a. [6 points] Find the gradient vector f at the point (1, 2). b. [6 points] Let g(1, 3, 5) = (1, 2). And g(1, 3, 5) = (2, 4, 2). If h(x, y, z) = f(g(x, y, z)) Find h(1, 3, 5). c. [6 points] Approximate by hand h(2, 3, 4). Briefly explain.

5 Math 116 / Exam 1 (October 7, 2014) page 5 3. [10 points] True or False, no partial credit, no explanation. a. [2 points] The line parametrized by p(t) = (t, t, t) intersects 4z = x 2 = y at the point (1, 1, 1). b. [2 points] For any vectors u and v in three dimensional space, u v = v u. c. [2 points] The crossproduct u v is the area of the parallelogram with edges u and v. d. [2 points] For any vectors u and v in three dimensional space, u (v u) = 0. e. [2 points] The arc length of a curve given by p(t) with a t b is given by b a p(t) dt.

6 Math 116 / Exam 1 (October 7, 2014) page 6 4. [8 points] Explain why two level curves of a function f(x, y) cannot intersect. 5. [5 points] Let b = (x, y, z) be any vector such that b is perpendicular to < 1, 3, 2 >. Solve for the equation defining all such b.

7 Math 116 / Exam 1 (October 7, 2014) page 7 6. [19 points] Given a point in polar coordinates (r, θ) there is a function f(r, θ) = (x, y) which gives us the rectilinear coordinates. The Martians have a slightly different way of describing vectors (for a full description, read The Martian Chronicles ). They use (a, b) which satisfy (x, y) = M(a, b) = (3a cos(b), 4a sin(b)). a. [5 points] Plot the rectilinear coordinate (1, 1) and convert to polar coordinates. b. [7 points] Sketch (with explanation and labeled points) the Martian equation a = 2. c. [7 points] Write an integral (but do not evaluate the integral) which will equal the length of the curve from part (b).

8 Math 116 / Exam 1 (October 7, 2014) page 8 7. [14 points] Here is the graph of a function f(x, y) = z: a. [7 points] What is the lim (x,y) (0,0) f(x, y)? Explain how you calculated it.

9 Math 116 / Exam 1 (October 7, 2014) page 9 b. [7 points] Circle the plot of the level curves of f(x, y). Briefly explain your choice.

10 Math 116 / Exam 1 (October 7, 2014) page [6 points] Let f(x, y) be a function, and let g(r, θ) = (x, y) be the standard function converting from polar to rectilinear coordinates. Calculate the θ derivative of the function h(r, θ) = f(g(r, θ)). Your answer should be in terms of f x and f y. 9. [2 points] What is your favorite novel or biography?