Consider the top half of the sphere x 2 y 2 z 2 9 a

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Exploring Microeconomics
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Chapter 10 / Exercise 5
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1. Consider thetop halfof the spherex2+y2+z2= 9.(a) Solve forzto findz=p9-x2-y2. Now find∂z∂xand use this formula to find∂z∂xwhenx= 1, y= 2, z= 2.(b) Redo part (a) by an alternate method: Use implicit differentiation to find∂z∂xifx2+y2+z2= 9 whenx= 1, y= 2, z= 2. Which method is easier?2. Findgz(x, y, z) ifg(x, y, z) =zx-y-yz2.
3. Can two different level curves for the same function intersect? In other words, ifc16=c2, is it possible for levelcurvesf(x, y) =c1andf(x, y) =c2to intersect? If so, give an example. If not, justify why it is impossible.
4. Iff(x, y, z) = ln(z+y2)·ln(x), findfz(x, y, z) andfzy(x, y, z).
5. We say a functionf(x, y) isharmoniciffxx(x, y) +fyy(x, y) = 0 for allxandy.Show that the functionf(x, y) = ln(x2+y2) is harmonic.(technical note:fis undefined at (0,0), but we still callfharmonic anyway)
6. (6 pts) Figure 11 on page 393 of our softcover textbook shows some level curves for the functionz=f(x, y).(a) What isf(2,3)? Solve the equationf(x,3) = 8 forx.(b) Find the smallest value ofz=f(x, y) ifx= 2. What is the corresponding value ofy?(c) What are the signs off01(x, y) andf02(x, y) at the pointsA, B, andC? Estimate the values of these partialderivatives atA.Solution
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Exploring Microeconomics
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Chapter 10 / Exercise 5
Exploring Microeconomics
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