MATH 200 EXAM 2 SPRING April 27, 2011
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1 MATH 00 EXAM SPRING 00-0 April 7, 0 Name: Section: ONLY THE CORRECT ANSWER AND ALL WORK USED TO REACH IT WILL EARN FULL CREDIT. Simplify all answers as much as possible unless eplicitly stated otherwise. This is a closed-book, closed-notes eam. No electronic devices are allowed. IF YOUR SECTION NUMBER IS MISSING OR INCORRECT, 5 POINTS WILL BE DEDUCTED FROM YOUR SCORE. Points PAGE points Score PAGE 0 points PAGE 3 0 points PAGE 4 0 points Points PAGE 5 0 points Score PAGE 6 0 points PAGE 7 8 points Total:
2 . (6 points each) Consider the vector-valued function r () t 3cos,sin t t. a. On the aes below sketch the graph of r () t. Indicate the direction of increasing t. b. Find r '( ) and sketch it on the aes below. For full credit the length of r '( ) must be correct. 3 y
3 EXAM MATH 00 SPRING 00. (6 points each) Consider the vector-valued function r () t cos, t 4sint. a. On the aes below sketch the graph of r () t. Indicate the direction of increasing t. b. Find r ' and sketch it on the aes below. For full credit the length of r ' must be correct. 5 y
4 r() t ti t j 3tk. (0 points) Find the point (or points) where the graph of intersects the plane 4 yz (0 points) Find the tangent line to the graph of () 9 r t t i t j t 4 kat the point 0,9,. Epress your answer as a vector equation or as a set of parametric equations.
5 EXAM MATH 00 SPRING 00. (0 points) Find the point (or points) where the graph of r() t 3ti4 t j(35) t k intersects the plane y3z (0 points) Find the tangent line to the graph of () 9 r t t i t j t 4 kat the point 0,9,. Epress your answer as a vector equation or as a set of parametric equations.
6 4. (0 Points) Solve the vector initial-value problem for r () t. r't () t i t j, r() i4j. 3
7 5. (0 points) On the aes below sketch the domain of f ( y, ) y. Use solid lines for any portions of the boundary included in the domain and dashed lines for any portions not included. 3 y
8 EXAM MATH 00 SPRING (0 points) On the aes below sketch the domain of f ( y, ) y. Use solid lines for any portions of the boundary included in the domain and dashed lines for any portions not included. 3 y
9 6. (0 points) Consider the surface y z. On the aes below sketch the traces of the surface in the three coordinate planes. Identify the surface. y-plane 3 y z-plane 3 z yz-plane 3 z y Surface Name: 5
10 7. (0 Points) Use the information in the following picture to find the values of the first-order partial derivatives of f at the point,. Note: The two lines shown in the picture are tangent to the surface z f(, y) at the point (,,4). 8. (0 points) The volume V of a right circular cone with radius r and slant height s is given by V r s r. If r has a constant value of 5 cm, find the instantaneous rate of 3 change of V with respect to s when s 0 cm. 6
11 9. (8 points) Find the mied second-order partial derivatives of the function f ( y, ) e siny. 0. (0 points) Calculate, y, and z. tan( z) z y z using implicit differentiation. Leave your answer in terms of 7
12 EXAM MATH 00 SPRING (8 points) Find the mied second-order partial derivatives of the function f ( y, ) e sin y. 0. (0 points) Calculate, y, and z. yz 3 5y tanz z y using implicit differentiation. Leave your answer in terms of 7
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