Math 11 Fall 2018 Midterm 1

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1 Math 11 Fall 2018 Midterm 1 October 3, 2018 NAME: SECTION (check one box): Section 1 (I. Petkova 10:10) Section 2 (M. Kobayashi 11:30) Section 3 (W. Lord 12:50) Section 4 (M. Kobayashi 1:10) Instructions: 1. Write your name legibly on this page, and indicate your section by checking the appropriate box. 2. Except on clearly indicated short answer problems, you must explain what you are doing, and show your work. You will be graded on your work, not just on your answer. Make it clear and legible so we can follow it. 3. It is fine to leave your answer in a form such as ln(.02) or 239 or (385)(13 3 ). However, if an expression can be easily simplified (such as e ln(.02) or cos(π) or (3 2)), you should simplify it. 4. This exam is closed book. You may not use notes, computing devices (calculators, computers, cell phones, etc.) or any other external resource. It is a violation of the honor code to give or receive help on this exam.

2 Problem Points Score Total 90 2 /??

3 1. [16 points] Please indicate whether the following statements are TRUE (T) or FALSE/NOT NECESSARILY TRUE (X). You do not need to justify your answer. X (T/X) If u v = 0, then either u = 0 or v = 0. T (T/X) For any vectors u and v, u (u v) = 0. X (T/X) The points A = (3, 3, 4), B = (1, 2, 1), and C = (7, 5, 9) lie on the same line. T (T/X) For any vectors u and v, proj u (u v) = 0. 3 /??

4 2. [15 points] Let L 1 be the line defined by the equations and let L 2 be the line defined by (a) Show that the lines are skew. x 1 2 x 2 = y 3 3 = y 3, z = 1, = z Need to show that the lines are not parallel and do not intersect. Not parallel: L 1 is parallel to v 1 = 2, 3, 0. L 2 parallel to v 2 = 1, 3, 2. Do not intersect: Assume the lines intersect at (x, y, z). Then by L 1, z = 1. Plugging this result into L 2 yields x = 2, y = 3, but (2, 3, 1) is not on L 1. (Could also have stated a parametrization of each line, r 1 (s), r 2 (t), and shown that there are no s and t such that r 1 (s) = r 2 (t).) (b) Find the equations of two parallel planes that contain the lines L 1 and L 2. Define v 1 = 2, 3, 0, v 2 = 1, 3, 2. A normal vector to each plane is n = v 1 v 2 = 6, 4, 3. Plane 1: Pick a vector with terminal point on L 1, such as p 1 = (1, 0, 1). An equation of the plane is n (r 1 p 1 ) = 0 6(x 1) 4y + 3(z + 1) = 0 6x 4y + 3z = 3. Plane 2: Pick a vector with terminal point on L 2, such as p 2 = (2, 3, 1). An equation of the plane is n (r 2 p 2 ) = 0 (1) 6(x 2) 4(y 3) + 3(z + 1) = 0 (2) 6x 4y + 3z = 3. (3) (c) What is the distance between the two planes? Using p 1, p 2 from part (b), let u = p 2 p 1 = 1, 3, 0. (4) A unit vector in the direction of the distance between the planes is n/ n = , 4, 3. Therefore the distance is comp n u = 2 u n/ n (5) = 6/ 61. (6) 4 /??

5 3. [12 points] Let A = (2, 1, 3), B = (7, 1, 4), C = (0, 2, 0) be points. (a) Find the area of the space triangle with vertices A, B, C. Choose point A as the common vertex of the two vectors AB, AC. Then AB = 5, 0, 1, and AC = 2, 3, 3. The area is half of the norm of the cross product. The cross product is Therefore, AB AC = 3, 13, 15. (7) Area = 1 2 AB AC (8) = 1 3, 13, 15 (9) 2 = (10) Choosing either B or C as the common vertex may introduce a sign into the cross product which will not affect the final answer. (b) Find the equation of the plane containing A, B, C. The normal vector in part (a) is n = 3, 13, 15 (up to a factor of ±1). Using point A, 3(x 2) + 13(y 1) 15(z 3) = 0, (11) or 3x + 13y 15z = 26. (12) 5 /??

6 4. [12 points] A particle moves in space with position vector r(t) = t, t 2, t 3. (a) Find the velocity of the particle at the point (1, 1, 1). Plugging in t = 1 yields (b) What is the acceleration of the particle at (1, 1, 1)? so that (c) Determine T(1), where T is the unit tangent vector for r. r (t) = 1, 2t, 3t 2. (13) r (1) = 1, 2, 3. (14) r (t) = 0, 2, 6t, (15) r (1) = 0, 2, 6 (16) T(1) = r (1) r (1) (17) = , 2, 3. (18) (d) At what time is the speed exactly 1? s = r (t) = 1, 2t, 3t 2 = 1 + 4t 2 + 9t 4. (19) Plugging in s = 1 gives the equation 1 = 1 + 4t 2 + 9t 4, which only has a (real) solution when t = 0. (e) Calculate the following integral: ˆ 1 1 r (t) dt. ˆ 1 1 ˆ 1 1, 2t, 3t 2 dt = 1 ˆ 1 ˆ 1 dt, 2t dt, 3t 2 dt 1 1 (20) = 2, 0, 2. (21) (f) Write the expression for the length of the curve from t = 1 to t = 1. Do not attempt to evaluate the expression! s(t) = ˆ 1 1 r (t) dt, or s(t) = ˆ t2 + 9t 4 dt (22) 6 /??

7 5. [8 points] Does the limit exist? Justify your answer. 2x 2 y 2 lim (x,y) (0,0) xye x Does not exist because of different limits along different paths. Along the path (x(t), y(t)) = (t, mt), for m 0, which depends on m. 2t 2 m 2 t 2 lim t 0 mt 2 e t = 2 m2 m, (23) 7 /??

8 Math 11 Midterm 1 6. [12 points] Let f (x, y) = October 3, 2018 x. x + y2 (a) Sketch the level curve corresponding to z = 1/2. x 1 = = y 2 = x. 2 2 x+y (24) This shows that the domain is included in y 2 = x. Since the denominator must not be 0, however, (x, y) = (0, 0) must be excluded from the domain. (b) Sketch the vertical trace corresponding to x = 1. If x = 1 then g(y) f (1, y) = A plot of g(y) should satisfy the following identifiability criteria: 1. 1+y 2 0 < g(y) 1 g(0) = 1, g 0 (0) = 0 lim y g(y) = 0. 8 /??

9 (c) Describe the domain of this function. Domain: x + y 2 0 x y 2 y ± x. In the following plot the domain is everything but the dotted line. (d) Find the equation for the plane tangent to f(x, y) at the point (0, 1, 0). A normal vector to the tangent plane is given by n = f x, f y, 1. The partials at x = 0, y = 1 are f x (0, 1) = 1, and f y (0, 1) = 0. Therefore n = 1, 0, 1, and using the point (0, 1, 0), the plane is defined by 1(x 0) + 0(y 1) 1(z 0) = 0 (25) x z = 0. (26) 9 /??

10 7. [15 points] Choose the correct contour plot (A through I) for each function described below. Note that the x-axis is horizontal, the y-axis is vertical, contour lines are drawn for equallyspaced values of f and slightly shaded so that darker lines have a lower value than lighter lines. (You do not need to show your work.) (a) H f(x, y) = sin(πx). (b) D f(x, y) has a horizontal tangent only at the origin. (c) B f(x, y) is a plane with normal vector 1, 1, 1. (d) E f(x, y) is a cone with circular base on the xy plane and apex (the pointy end) with positive z coordinate. (e) I f(x, y) has f x = 0 for all values (x, y). A B C D E F G H I 10 /??

Math 22 Fall 2018 Midterm 2

Math 22 Fall 2018 Midterm 2 Math 22 Fall 218 Midterm 2 October 23, 218 NAME: SECTION (check one box): Section 1 (S. Allen 12:5) Section 2 (A. Babei 2:1) Instructions: 1. Write your name legibly on this page, and indicate your section