Math 20E Practice Midterms

# Math 20E Practice Midterms - ∇ f ◦ g a where a ∈ R m...

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Practice Midterm Examination Instructor J. Verstraete Time: 40 minutes No notes allowed All questions carry equal weight Question 1. State precisely the ² - δ deﬁnition of lim x a f ( x ) = L for a function f : R n R . Then prove using the ² - δ deﬁnition of limits that lim ( x,y ) (0 , 0) sin( x 2 + y 2 ) = 0 . 1

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Question 2. Find the direction of steepest increase of the function f ( x,y ) = ( x + y ) e xy from the origin. What is the equation of the tangent hyperplane to the surface z = f ( x,y ) at the origin? 2
Question 3. State precisely the chain rule for determining the gradient of

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Unformatted text preview: ∇ ( f ◦ g )( a ) where a ∈ R m and f : R n → R p and g : R m → R n are functions. Then determine ∇ ( f ◦ g )(1 , 1) when f : R 2 → R 3 is deﬁned by f ( x,y ) = ( x,y,xy ) and when g : R → R 2 is deﬁned by g ( z ) = ( z, 1 /z ). 3 Question 4. Find all second order partial derivatives for the function f : R 3 → R deﬁned by f ( x,y,z ) = (1 + x )(1 + y )(1 + z ) . 4...
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Math 20E Practice Midterms - ∇ f ◦ g a where a ∈ R m...

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