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213-zhu-07faex02

213-zhu-07faex02 - MATH 213 MIDTERM2(FALL 2007 Name Section...

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Unformatted text preview: MATH 213: MIDTERM2 (FALL 2007) Name:__,_; _____________________________ __ Section: _________ __ TA: ________________ _- Score: Problem 1 __________________________ __ Problem ________ _;___; _____ "I _____ __ Problem 3 ___________________________ __ Problem 4.__-_____.__~______;___-_-_ Problem 5.___-__-_-___r__-___-___--__ Problem 6.____ ____ _; ________________ __ I Total:r___;_;_____~____________-_ Instruction: Show all work. No work 2 no credit, even if you have a correct answer. References and calculator are not allowed. Problem 1 (15 points): Consider the function z 2 f (cc, y) : x2y3. (a) (10 points) Compute the partial derivatives g and g5. Then write down the total differential dz. ' ' (b) (5 points) Use the total differential to ﬁnd the approximate value of (2.01)2(0.99)3 Problem 2 (10 points): Let f (ac, y) = 2x3 +fy2 — Gary +27 Find all its critical points. Determine if they are'points for relative maximum, relative minimum or saddle points. ' Problem 3 (15 points): When a factory buys a: A machines and y B ma— chines the‘productivity function is P(x, 3/) 2 3023;“. Machine A costs 3 thou- sand dollars each and machine B costs 2 thousand dollars each. Use La- grange’s method to ﬁnd the combination of A machines and B machines the factory can buy for 54 thousand dollars which will maximize the productivity function. Problem 4 (10 points): Evaluate the double integral / /- exsdxdy, R Where R is the region in the xy-plane bounded by x = 1, y = 0 and y = \$2. ( Hint: choose a good order of integration. ) Problem 5 (10 points) Solve the following initial value problem: Problem 6 (10 points): Consider the initialvalue problem- dy ’ . —— 2 1 — 2 ' 0 = 1'. Use Euler’s method with stepsize h = 0.1 to ﬁnd the approximate values of I the solution at a: r: 0.1, 0.2 and 0.3. ...
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