Tuesday, September 18 ∗∗ Partial derivatives and differentiability.1. Consider f (x, y) = x y2ex2. Compute the following:∂ f∂x∂ f∂y∂2f∂x∂y∂2f∂y∂xWhat is the relationship between your answers for the last two? This is an instance of Clairaut’sTheorem and holds for most functions.2. Shown are some level curves for the function f : R2→ R. Determine whether the followingpartial derivatives are positive, negative, or zero at the point P :fxfyfxxfy yfx y= fyx3. The wind-chill index W = f (T, v) is the perceived temperature when the actual temperature isT and the wind speed is v. Here is a table of values for W .(a) Use the table to estimate∂ f∂Tand∂ f∂vat (T, v) = (−20,40).(b) Use your answer in (a) to write down the linear approximation to f at (−20,40).(c) Use your answer in (b) to approximate f (−22, 45).4. Consider f (x, y) =p1 −x2−y2.(a) What is the domain of f ? That is, for which (x, y) does the function make sense?(b) Describe geometrically the sur face which is the graph of f .(c) Find the tangent plane to the graph at (x, y) = (1/2, 1/2).(d) Consider the vector v =¡1/2,1/2,1/p2¢which goes from 0 to the point on the graph wherewe just found the tangent plane. What is the angle between v and a normal vector to thetangent plane?5. Consider f (x, y) =3px3+y3.(a) Compute fx(0,0). Note: this partial derivative exists.(b) Is f differentiable at
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