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20e-pm3

# 20e-pm3 - ∂x in terms of derivatives of f with respect to...

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Practice Midterm Examination Instructor J. Verstraete Time: 40 minutes No notes allowed All questions carry equal weight Question 1. Prove that lim ( x,y ) (0 , 0) sin( xy ) x 2 + y 2 does not exist. 1

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Question 2. State the deﬁnition of diﬀerentiability of a function f : R n R . Find the derivatives f x (0 , 0) and f y (0 , 0) of the function f ( x,y ) = x 1 / 3 y 1 / 3 . State without proof whether this function is diﬀerentiable at (0 , 0). 2
Question 3. Let f : R 2 R 2 be a diﬀerentiable function where f ( x,y ) = ( u ( x,y ) ,v ( x,y )). Let φ ( x,y ) = f ( f ( x,y )). Determine a formula for ∂φ

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Unformatted text preview: ∂x in terms of derivatives of f with respect to u and v and derivatives of u and v with respect to x and y . 3 Question 4. Let f : R 3 → R be a function. How many diﬀerent second order partial derivatives can f have? Now suppose f ∈ C 2 ( R 3 ). How many diﬀerent second order partial derivatives can f have? Find all second order partial derivatives of the function f : R 3 → R deﬁned by f ( x,y,z ) = x y + y z + z x . 4...
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20e-pm3 - ∂x in terms of derivatives of f with respect to...

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