Question

Calculus III

1) Identify each of the following surfaces: а) z' %3x? - 5у" b) z 4x2-4 y c) z2+3x2-5y = 4 d) z2x23-5y e) у3х* 2) Find and classify all of the critical points for f(x, y)=xy -x2 + y'. 3) Find the maximum and minimum values of f(x,y)=xy over the ellipse х* + 2y %3D1. 4) Let fx, y) x3 -cosy a) Find the first order Taylor polynomial for f(x,y) based at (1,7). b) Find the sccond order Taylor polynomial for f(x,y) based at (1,n). c) Roughly, what will the graph of f(x.y) look like near the point (1, zn )? 5) Compute all the first order partial derivatives for each of the given functions: а) f(х, у)%3D sin(xу") х'z b) g(x. y,z) Зу +2 2) Let f(x, y) x'y -4yx+y2. a) Find Vf(2,6) b) Find the directional derivative of f(x, y) at (2,6)as you head towards the point (9,1) c) In which direction is f(x, y) decreasing most rapidly at (4,5)? What is the maximum rate of change? 3) Find all the second order partial derivatives for f(x, y)= x2y х-у 4) Find the domain of the function f(x, y)= Sketch the domain in the ху — plane. 5) The temperature of a gas at the point (x, y, z) is G(x, y,z) sinh(x)cos(y)sin(z). a) What is the rate of changc of the tempcrature at (0,, -) in the dircction 4 2 31-2j+k? b) At (1,2, 3), which direction would you be facing if you were facing in the direction of greatest increase in temperature? 6) The contour map of f(x, y) is given a) At point A, is f^ positive, negative, or zero? Explain. b) Sketch the gradient off at point A.

Answer 1

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