Tuesday September 18 Partial derivatives and differentiability 2 1 Consider f x y x y 2 e x Compute the following f y f x 2 f x y 2 f y x What is the relationship between your answers for the last two This is an instance of Clairaut s Theorem and holds for most functions 2 Shown are some level curves for the function f R2 R Determine whether the following partial derivatives are positive negative or zero at the point P fx fy f xx fyy fx y f yx 3 The wind chill index W f T v is the perceived temperature when the actual temperature is T and the wind speed is v Here is a table of values for W a Use the table to estimate f T and f v at 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 p 1 x 2 y 2 a What is the domain of f That is for which x y does the function make sense b Describe geometrically the surface which is the graph of f c Find the tangent plane to the graph at x y 1 2 1 2 p d Consider the vector v 1 2 1 2 1 2 which goes from 0 to the point on the graph where we just found the tangent plane What is the angle between v and a normal vector to the tangent plane p 5 Consider f x y 3 x 3 y 3 a Compute f x 0 0 Note this partial derivative exists b Is f differentiable at 0 0
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