Calculus assignment

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timer Asked: Mar 18th, 2018

Question Description

Solve all questions step-by-step since i need to use all solutions for my exam revision . My lecturer also needs to see how i did get the answers . Please make sure all solutions are correct and concise

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241 Exam Final Show all work. No calculators allowed. Lawrence Sze Name June 10 2011 No Calculators allowed. You may assume the following formulae: If P x, y, z P r, , z , then y sin x r cos r x2 y2 2 x y2 y If P x, y, z P , , sin cos x2 y sin sin y tan −1 x z cos cos −1 P , , sin z cos D D x 2 y2 y2 z2 y sin x2 x cos z y2 x2 x 2 y2 x2 sin z2 y2 x2 y2 y2 z2 1 m D then r x, y dA x cos , then x If P r, , z m y tan −1 x r sin r2 z cos D a, b x z2 r 2 z 2 sin f xx a, b f yy a, b − f xy a, b 1 m D x x, y dA r r 2 z2 2 y y x, y dA 1. (5 points) Describe in words the surface given by ∂z ∂z and ∂y if 2. (10 points) Find ∂x ln x yz 3. (15 points) Let F 〈2xz y 2 , 2xy, x 2 3z 2 . a. Find a function f such that F ∇f. b. Evaluate F dr along the curve C : x C 1 2 sin 2 cos 2 cos 2 4 xy 2 z 3 t 241 , y 1 t 241 , z 1 t 241 − t 241 0 ≤ t ≤ 1. 4. (10 points) Find the limit if it exists, or show that the limit does not exist. x3 y3 lim x,y → 0,0 x 2 y2 5. (15 points) Find the equation of the the tangent plane to the surface z 1, 0, 0 6. (9 points) Match the graph with it’s equation A______________ 1 xe y cos z at B___________________ C_________________ a. x 2 4y 2 9z 2 1 b. y 2x 2 z 2 c. x 2 − y 2 z 2 1 d. −x 2 y 2 − z 2 1 e. x 2 2z 2 1. 7. (20 points) First, compute y 3 dx − x 3 dy directly, and then by applying Green’s Theorem C where C is the positively oriented circle x 2 y 2 4. 8. (15 points) Why is the line integral 2x sin ydx x 2 cos y − 3y 2 dy independent of path? C Also, evaluate the integral where C is any path from −1, 0 to 5, 1 . 9. (20 points) Find the mass and the center of mass of E where E is the solid that lies between the spheres x 2 y 2 z 2 1 and x 2 y 2 z 2 4 in the first octant. Assume the solid is constructed of a material with uniform density 1. 10. (15 points) Find the maximum and minimum values of the function f x, y, z 2x 6y 10z subject to the constraint x 2 y 2 z 2 35 11. (15 points) Evaluate the integral used to evaluate xydA, where R is the region R bounded by xy 1, xy 3, y x, and y 12x obtained by transforming the region R to a rectangular region using an appropriate change of variables. 12. Evaluate 5, 0 , x R 5 5 , 2 2 1 13. Evaluate 0 ydA where R is the trapezoidal region with vertices given by 0, 0 , and 1 5 2 1 x y3 1 , − 52 . dy dx
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