Hello, I am currently taking calculus 3, multivariable calculus. I was sick for over a week and missed classes. I did learn the material on my own at home through the textbook, but I am still not very confident in what I know. We were given practice questions for our upcoming midterm, so I would like for someone to do these 6 questions from so I can compare my answers with and see if I am on the right track. Please show all steps, and perhaps a quick explanation (quick!) about why you might be doing something, unless it should be obvious. The topics we have learned so far in this multivariable calculus 3 course are: functions of several variables, limits and continuity, partial derivatives, tangent planes, linear approximations, chain rule, directional derivatives, gradient vector, min and max values, Lagrange multipliers, double integrals over rectangles, double integrals in general regions and in polar coordinates Q#1. The temperature at a point (x,y,z) is given by T(x,y,z) = 200e?x2?3y2?9z2 , where T is measured in in C and x, y, z in meters. (a) Find the rate of change of the temperature at the point P (2, ?1, 2) in the direction toward the point (3, ?3, 3). (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P . Q#2. Find three positive numbers x,y,z whose sum is 100 such that xaybzc is a maximum. Q#3. Of all triangles of a given perimeter 2p, find the one that has the largest area. Hint: use Heron?s formula and Lagrange multipliers. Q#4. Use Lagrange multipliers to find extreme value(s) of the function f(x1,x2,…,xn)= x1 +x2 +???+xn subject to x12 +x22 +???+xn2 =1. Q#5. Calculate the double integrals (see attached file (pic) Q5) Q#6. Calculate a bounded double integral (see attached file (pic) Q6) Attachment 1 Attachment 2 ATTACHMENT PREVIEW Download attachment // Ensure that we’re not replacing any onload events function addLoadEvent(func) { var oldonload = window.onload; if (typeof window.onload != ‘function’) { window.onload = func; } else { window.onload = function() { if (oldonload) { oldonload(); } func(); } } } addLoadEvent(function(){load1();}); var isIE = false; function load1(){ } ATTACHMENT PREVIEW Download attachment // Ensure that we’re not replacing any onload events function addLoadEvent(func) { var oldonload = window.onload; if (typeof window.onload != ‘function’) { window.onload = func; } else { window.onload = function() { if (oldonload) { oldonload(); } func(); } } } addLoadEvent(function(){load1();}); var isIE = false; function load1(){ }

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