Math - Final Exam - 6/13/13 NAME: SECTION: Directions: For the free response section, you must show all work. Answers without proper justification will not receive full credit. Partial credit will be awarded for significant progress towards the correct answer. Cross o any work that you do not want graded. You have two () hours to complete this exam. When time is called, STOP WRITING IMMEDIATELY. You may not use any electronic devices including (but not limited to) calculators, cell phones, or ipods. Problem 1 Problem Problem 3 Problem 4 Multiple Choice TOTAL 1 Points 1 Points 1 Points 1 Points 5 Points 1 Points Section Class Times Instructor Section Class Times Instructor 1 9: am 9:5 am Huilan Li 13 1: pm 1:5 pm Dimitrios Papadopoulos 11: am 11:5 am Jason Scott Aran 14 : pm :5 pm Jason Scott Aran 3 11: am 11:5 am Dennis Yang 15 9: am 9:5 am Hwan Yong Lee 4 4: pm 4:5 pm Dennis Yang 16 1: pm 1:5 pm Daryl Lawrence Falco 6 9: am 9:5 am Daryl Lawrence Falco 17 4: pm 4:5 pm Alexander Dolgopolsky 7 1: am 1:5 am Harold D Gilman 18 1: pm 1:5 pm Jason Scott Aran 8 1: am 1:5 am Hwan Yong Lee 19 1: am 1:5 am Daryl Lawrence Falco 1 : pm :5 pm Alexander Dolgopolsky 1: pm 1:5 pm Alexander Dolgopolsky 11 1: pm 1:5 pm Dimitrios Papadopoulos 1 5: pm 5:5 pm Dennis Yang 1 4: pm 4:5 pm Dimitrios Papadopoulos 1
Free Response For each of the following problems, you must show all of your work to earn full credit. 1. Find the acute angle between the planes which are tangent to surfaces S 1 : x +y =5 and S : z = x at the point (3, 4, 1). (You may leave your answer in terms of an inverse trigonometric function.)
. Evaluate the following iterated integral by converting to polar coordinates. Z Z p 16 x 1 dy dx p 3x 1+x + y 3
3. Find all absolute extrema of f(x, y) =x xy +y on the closed rectangular region shown below. 4
4. The solid shown below is enclosed by z =,z =,y = x, y = x, andtheyz-plane. (a) Set up a triple integal (or triple integrals) in rectangular coordinates with the order of integration as dz dy dx which represents the volume of the solid. (b) Set up a triple integal (or triple integrals) in rectangular coordinates with the order of integration as dx dz dy which represents the volume of the solid. Do not evaluate any of the integrals. 5
Multiple Choice Circle the letter of the best answer. Make sure your circles inlude just one letter. These problems will be marked as correct or incorrect; partial credit will not be awarded for problems in this section. Each problem is worth 4 points. 5. Suppose f(x, y) is a twice di erentiable function which satisfies the following table of values: (x, y) f(x, y) f x (x, y) f y (x, y) f xx (x, y) f yy (x, y) f xy (x, y) (, ) 9 1 1 (1, ) 3 1 (, 6) 5 3 1 Which of the following statements is true? I. f(x, y) hasalocalminimumat(x, y) =(, ) II. f(x, y) has a critical point at (x, y) =(1, ) III. f(x, y) hasasaddlepointat(x, y) =(, 6) (a) Ionly (b) III only (c) IandIIonly (d) IandIIIonly (e) II and III only 6
6. Which of the following is the double integral that results from reversing the order of integration on Z 3 Z p x+1 1 f(x, y) dy dx? (a) (b) (c) (d) (e) Z 3 Z y +1 1 1 Z Z y 1 Z Z y 1 1 Z Z 1 y 1 Z Z 3 1 y 1 f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy f(x, y) dx dy 7. Consider the line L which contains the point A(1,, 1) and is perpendicular to the plane P : x y + z =1. AtwhichpointwilllineL intersect plane P? (a) (, 1, ) (b) (1, 1, 1) (c) (, 3, ) 4 (d) 3, 5 3, 4 3 (e) 3, 1 3, 3 7
8. Which of the following vectors is normal to the plane determined by points A(, 1, ), B(, 1, 1), and C(,, 1)? (a) h, 1, i (b) h1,, 4i (c) h1, (d) h1, 3, (e) h1, 6, i i 1, 4i 9. What is the largest rate of increase of f(x, y) =ln(x + y) atp (, 1)? (a) (b) 1 (c) p (d) p 3 (e) ln 8
1. Which of the following is the Jacobian of the transformation x = e u cos v, y = e u sin v? (a) e u (b) e u (c) e u cos v (d) e u cos v (e) e u cos v e u sin v e u sin v e u sin v 11. What is the distance from P (3, 3, 4) to the x-axis? (a) 3 (b) 4 (c) 5 (d) p 9 (e) p 34 9
1. Let R be the region in the xy-plane shown above. Using the transformation x = 1 (u+v) and y = 1 ZZ (u v), which of the following integrals is equivalent to (x y)e x+y da? R HINT: If x = 1 (u + v) andy = 1 (u v), then u = x + y and v = x y. (a) (b) Z 4 Z 4 1 Z 1 Z 4 1 ve u dv du ve u dv du (c) (d) 1 4 (e) 1 Z 4 Z 3 1 Z 4 Z 4 1 Z 4 Z 4 1 ve u dv du ve u dv du ve u dv du 1
The following two questions refer to: Z 3 Z p 9 x Z p 9 x y x dz dy dx 13. Which of the following triple integrals in Cylindrical Coordinates is equivalent to Z 3 Z p 9 x Z p 9 x y x dz dy dx? (a) Z 3 Z p 9 r r cos dz dr d (b) Z p 9 r Z 3 r cos dz dr d (c) Z 3 Z p 9 r r 3 cos dz dr d (d) Z 3 Z p 9 r r 3 cos dz dr d (e) Z p 9 r Z 3 r 3 cos dz dr d 14. Which of the following triple integrals in Spherical Coordinates is equivalent to Z 3 Z p 9 x Z p 9 x y x dz dy dx? (a) (b) (c) (d) (e) 4 Z 3 Z 9 Z 3 Z 3 Z 9 cos sin cos sin 4 cos sin 4 cos sin 3 4 cos sin 3 d d d d d d d d d d d d d d d 11
15. Consider the following initial value problem. 8 < r (t) =he t,, i : r() = h, 3, 4i Which of the following is r(1)? (a) r(1) = e +1, 1, 4 (b) r(1) = e +1, 1, (c) r(1) = e +1,, 4 (d) r(1) = e,, (e) r(1) = e,, 4 16. Suppose v and w are arbitrary non-zero vectors and let k be an arbitrary scalar. Which of the following statements is FALSE? (a) v w = v w (b) v w = (w v) (c) v (v w) = (d) v (kv) = (e) kkvk = k kvk 1
17. Some level curves of a function f(x, y) areshownbelow. Which of the following statements is true about @f @x (a) @f @x > and@f @y > (b) @f @x > and@f @y < (c) @f @x < and@f @y > (d) @f @x < and@f @y < and @f @y at the point P? (e) There is not enough information to determine the signs of these partial derivatives. 13