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213-Zhu-07fafin - MATH 213 FINAL(FALL 2007 Score V Problem...

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Unformatted text preview: MATH 213: FINAL (FALL 2007) Score: _ V . Problem 1’._-_______-»~____; ___________ Problem 2 ____________________________ Problem 3 ____________________________ Problem 4 ____________________________ Problem 5f;___.;_,_-_..__-______c _ . Problem 6.____r_w____; _______________ Problem 7 ____________________________ Total: ___________________________ Instruction: Show all work. No work = no credit, even if you have a correct answer. References and calculator are not allowed. Problem 1 (25 points): Find each integral: (a) (5 points) f(a: + %)2d$ (b) (5 points) fol fiez‘fldm (c) (5 points)fac21nxda: . ' + (d) (10 pOIHtS) f1 00de Problem 2 (10 points): Let 2 = f(a:,y) = 1 1+6? ‘ (a) (5 points) Compute the partial derivatives gg and 53-5. . .' ‘ . - 1 (b) (5 pomts) erte down dz. Then find the approx1mate value ()fo- Problem 3 (10 points): Find the maximum of f(:z;, y) : 332 — 43/2 subject to the VcOnstraint 393 + 4y = 9. Problem 4 (15 points): Evaluate the following double integrals: (a) (5 points) f fR efdxdy, where R is the region in lthe say-plane bounded by1£y£2and03x3y . (b) (10 points) fol f; \/ 51:2 + 1dasdy. (Hint: change the order of integration.) Problem 5: (a)(7 points) Find the Taylor polynomial of degree 2 for flan) : 8952+?“ at :6 = 0. (b) (8 points) Find the Taylor series for 222 at :6 = 0. What’s its interval _ of convergence? V 1- _ . - ‘ ' . , . Problem 6 (10 points): Mike is digging a well according to the following plan: In the first day he digs 10cm7 the second day he continues to dig 9cm, the third day 8.10m,...and the n—th day 10(0.9)"‘1cm. (a) (5 points) Find the the depth of the well in n days. (b) '(5 points) Suppose the "water is '10'1crn under theground Gan Mike finally dig out the water someday, according to .his' plan? Justify your answer. Problem 7 (15 points): Using L’Hospitai’s Rule to find .thefollowing limits: (a) (5 points) limb0 1&2ch (b) (10 points) limaHl (1n ...
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