pope2 - v = <2, -3, 4> 7. Use...

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Math 22 Exam 2 Name: _________________ Fall 2000 (please print) 1. Find the domain of the function f(x,y) = ln(x 2 + y 2 ) and sketch a contour map of f showing multiple level curves. 2 Show that does not exist. 3. Given find f x , f y , f z , and f xx 4. Find the linear approximation of f(x,y)=e x + sin(y) at (0,0) and use this to estimate e 0.1 + sin(-0.1) 5. Given f(x,y)=sin(x)cos(y), x(s,t)=s tan (t), y(s,t)=t 2 s 3 find Put your answer in terms of s and t only. 6. Given f(x,y,z) = xyz + x 2 + y 2 a)find the gradient of f b)find the direction of maximum increase of f at (1,1,1) c)find the maximum rate of change of f at (1,1,1) d)find the directional derivative of f at (1,1,1) in the direction of
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Unformatted text preview: v = <2, -3, 4> 7. Use LaGrange Multipliers to find the maximum and minimum values of f(x,y)=x 2 + 2y 2 when x 2 + y 2 = 1. 8. Find the absolute maximum and absolute minimum of f(x,y)=x 2 + 2y 2 on the closed, bounded region D={(x,y)| x 2 + y 2 1} (you may use your results from problem 7) 9. Use Lagrange Multipliers to set up (but do not evaluate) a system of 5 equations in 5 unknowns to maximize f(x,y,z)=sin(x) + cos(y) + tan(z) subject to the constraints x 2 + y 2 + z 2 = 1 and ye xz = 8 ) ( tan ) , , ( 1 z y x z y x f-= t f s f 2 2 2 ) , ( ) , ( ) ( lim y x y x y x + +...
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