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1 MTH 34 Solutions to Exam April 9th, 8 Name: Section: Recitation Instructor: INSTRUTIONS Fill in your name, etc. on this first page. Without fully opening the exam, check that you have pages through. Show all your work on the standard response questions. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip limits or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. You will be given exactly 9 minutes for this exam. Remove and utilize the formula sheet provided to you at the end of this exam. AADEMI HONESTY Do not open the exam booklet until you are instructed to do so. Do not seek or obtain any kind of help from anyone to answer questions on this exam. If you have questions, consult only the proctor(s). Books, notes, calculators, phones, or any other electronic devices are not allowed on the exam. Students should store them in their backpacks. No scratch paper is permitted. If you need more room use the back of a page for scratch work. Work on the back of pages will not be graded. Anyone who violates these instructions will have committed an act of academic dishonesty. Penalties for academic dishonesty can be very severe. All cases of academic dishonesty will be reported immediately to the Dean of Undergraduate Studies and added to the student s academic record. I have read and understand the above instructions and statements regarding academic honesty:. SIGNATURE Page of

2 MTH 34 Solutions to Exam April 9th, 8 Standard Response Questions. Show all work to receive credit. Please BOX your final answer.. (4 points) onsider the function f(x, y) x 4xy + y 4 + (a) Find the critical points of f and classify them as local minima, local maxima, or saddle points. Solution: f x 4x 4y f y 4x + 4y 3 which yield critical points at (, ), (, ), and (, ). at (, ) f xx 4 f xy 4 f yy y D (4)() ( 4) 6 < so (, ) is a saddle point. at (, ) D (4)() ( 4) 3 > and f xx 4so (, ) is a local min. at (, ) D (4)() ( 4) 3 > and f xx 4so (, ) is a local min. (b) Let be the curve y x for x [, 3]. What is the absolute minimum and maximum of f on? Solution: f(x, x) x 4x + x 4 + x 4 x + and so f (x) 4x 3 4x which has critical points at x and x on the interval x [, 3]. We have f() f() + f(3) So is the absolute min of f on and 65 is the absolute max of f on. Page of

3 MTH 34 Solutions to Exam April 9th, 8. (4 points) Evaluate ( (x + y)i + (y z)j + z k ) dr where is given by vector function r(t) t i + t 3 j + t k, t. Solution: r (t) t, 3t, t and so ( (x + y)i + (y z)j + z k ) dr x + y, y z, z t, 3t, t dt t + t 3, t 3 t, (t ) t, 3t, t dt (t 3 + t 4 ) + (3t 5 3t 4 ) + (t 5 ) dt 5t 5 t 4 + t 3 dt [ 5 6 t6 5 t5 + t4 ] Page 3 of

4 MTH 34 Solutions to Exam April 9th, 8 3. (4 points) Find the surface area of the surface given by the graph z + 3x + ln ( y ) 5 y over the domain x 4, y. Solution: f x 3 f y y y 5y y 5y y y and so the surface area can be given by 4 ( ) 4 (3) + + dy dx y y y + 4y ( ) y +y 4 dy dx 4y ( ) y +y 4 dy dx 4y ( ) +y +y 4 dy dx 4y ( ) + y dy dx y ( ) dx y + 5y dy [x] 4 [ ln y + 5 y] 3 [ ln + 5 (3)] 3 45 ln + 3 (ln + 5) Page 4 of

5 MTH 34 Solutions to Exam April 9th, 8 4. (4 points) onsider the vector field F(x, y, z) ( xy z 3) i + ( x yz 3) j + ( 3x y z ) k (a) ompute the curl(f) (show your calculations) Solution: i j k x y z xy z 3 x yz 3 3x y z (6x yz 6x yz )i (6xy z 6xy z )j + (4xyz 3 4xyz 3 )k,, (b) Find a function f such that F f. Solution: f xy z 3 dx f x y z 3 + g(y, z) f y x yz 3 + g y (y, z) x yz 3 so g y giving g(y, z) h(z) f z 3x y z + h (z) 3x y z and so h (z) giving us h(z) K so a possible final solution is f x y z 3. (c) Evaluate a line integral F T ds where is the curve shown below. z (, 3, 3) Solution: F T ds f(, 3, 3) f(,, ) (4)(9)(7) ()()( 8) x > > y > (,, ) (, 3, ) Page 5 of

6 MTH 34 Solutions to Exam April 9th, 8 Multiple hoice. ircle the best answer. No work needed. No partial credit available. 5. (4 points) E is the cube given by x a, y a, z a with mass density σ(x, y, z) x + y + z. The mass is given by the formula: a a x y A. dz dy dx B.. D. a a a a a a x y x + y + z dz dy dx dz dy dx a a E. None of the above. x + y + z dz dy dx 6. (4 points) Which of the following is the equations of the tangent plane to the surface xyz 6 at the point (3,, ). A. x + y 3 B. x + 3y + y. x 3y + 6y 4 D. x + 3y + z 4 E. y + 3z 4 7. (4 points) Evaluate A. 3π B. π. 5π D. π E. π B dv, where B lies between the spheres x + y x + y + z 4 and x + y + z 9 Page 6 of

7 MTH 34 Solutions to Exam April 9th, 8 8. (4 points) Reversing the order of integration A. B.. D. E. 6 y 5 5 3x + y dy dx (9 x)/ 5 x y 4 3x + y dy dx 3x + y dy dx 3x + y dy dx 3x + y dy dx 4 9 y 9. (4 points) If F(x, y, z) zi + (x + z)j + 3k, then div(f) A. B. i + j + 3k. + x + y D. xz E. x + y + z 3x + y dx dy. (4 points) Which of the following could be the gradient vector field f of the function f(x, y) xy. A. y B. y. y x x x D. y E. y x x Page 7 of

8 MTH 34 Solutions to Exam April 9th, 8. (4 points) Evaluate y, y dr, where is the line segment from (, ) to (, ). A. 3 B D. E. 9. (4 points) The area of the surface with parametric equations x u, y uv, z v, with u and v A. B.. 3 D. 4 E (4 points) onsider the closed curve which is the union of three paths; the straight line from (, ) to (, ), the arc of the unit circle from (, ) to (, ), and finally, straight line from (, ) to (, ). Evaluate (x + y)dx (x y)dy A. B. 3. π/ D. π/4 E. π Page 8 of

9 MTH 34 Solutions to Exam April 9th, 8 More hallenging Problem(s). Show all work to receive credit. 4. (8 points) onsider the vector field F x y x + y i + Ax + y x + y j (a) ompute the value of A so that the vector field is conservative on its domain. Solution: Q x A(x + y ) (Ax + y)(x) (x + y ) P y ( )(x + y ) (x y)(y) (x + y ) So since we want Q x P y we get Which yields A. (b) For this value of A, explain the equality Ax + Ay Ax xy x y xy + y Ay Ax x + y F dr at the origin and S is the boundary of the square [, ] [, ]. S F dr where is the unit circle centered Solution: onsider the region D inside the square and outside the circle on which F is continuous. Green s Theorem we have F dr F dr GT Q x P y da da S D and so F dr F dr S Using 5. (6 points) Let the vector field F(x, y) in the plane satisfies relation F(x, y) F( x, y). Show that the line integral F dr where is the boundary of the square [, ] [, ]. Solution: onsider the following definitions: F P, Q. L(t), t, t [, ], R(t), t, t [, ], B(t) t,, t [, ], T(t) t,, t [, ] which parametrize the sides of the square. Then F dr P, Q dr + P, Q dt + R P (, t), Q(, t), dt + + T B P (t, ), Q(t, ), dt + Q(, t) dt + Q(, t) dt P (t, ) dt + P (t, ) dt + P, Q db + P, Q dl L P (t, ), Q(t, ), dt P (t, ) dt + P (t, ) dt P (, t), Q(, t), dt Q(, t) dt Q(, t) dt Page 9 of

10 MTH 34 Solutions to Exam April 9th, 8 ongratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID ready so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 6 No more than points may be earned on the exam. Page of

11 MTH 34 Solutions to Exam April 9th, 8 FORMULA SHEET PAGE Vectors in Space urves and Planes in Space Suppose u u, u, u 3 and v v, v, v 3 : Line parallel to v: r(t) r + tv Unit Vectors: Length of vector u Dot Product: ross Product: i,, j,, k,, u u + u + u 3 u v u v + u v + u 3 v 3 u v cos θ u v Vector Projection: i j k u u u 3 v v v 3 Partial Derivatives proj u v u v u hain Rule: Suppose z f(x, y) and x g(t) and y h(t) are all differentiable then dz dt f dx x dt + f dy y dt u Plane normal to n a, b, c : a(x x ) + b(y y ) + c(z z ) Arc Length of curve r(t) for t [a, b]. L b a r (t) dt Unit Tangent Vector of curve r(t) T(t) r (t) r (t) More on Surfaces Directional Derivative: D u f(x, y) f u Second Derivative Test Suppose f x (a, b) and f y (a, b). Let D f xx (a, b)f yy (a, b) [f xy (a, b)] (a) If D > and f xx (a, b) >, then f(a, b) is a local minimum. (b) If D > and f xx (a, b) <, then f(a, b) is a local maximum. (c) If D < then f(a, b) is a saddle point. Geometry / Trigonometry Area of an ellipse x a + y is A πab b sin x ( cos x) cos x ( + cos x) sin(x) sin x cos x Page of

12 MTH 34 Solutions to Exam April 9th, 8 FORMULA SHEET PAGE Multiple Integrals Area: A(D) da D Volume: V (E) Transformations D E dv Polar/ylindrical r x + y x r cos θ y r sin θ y/x tan θ f(x, y) da f(r cos θ, r sin θ) r dr dθ f(x, y, z) dv E D f(r cos θ, r sin θ, z) r dz dr dθ E Spherical Additional Definitions curl(f) F div(f) F F is conservative if curl(f) Line Integrals Fundamental Theorem of Line Integrals f dr f(r(b)) f(r(a)) Green s Theorem P dx + Q dy (Q x P y ) da D Transformations x ρ sin φ cos θ y ρ sin φ sin θ z ρ cos φ ρ x + y + z E E f(x, y, z) dv f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)(ρ sin φ) dρ dφ dθ Page of