Calculus Homework

MAT 17 C - Luis Rademacher Midterm 1 - April 24th

50 minutes

Open book and open notes but every student must write his or her own solution. Justify

or prove all your answers.

1. Consider a real-valued function given by f(x, y) = p x + y. Find the largest possible

domain of f and sketch the level curve of f at levels 0 and 1.

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2. Consider the function given by f(x, y) = ⇣

y2�2x2 x2+y2

⌘3 .

Show that

lim (x,y)!(0,0)

f(x, y)

does not exist by computing the limit along the positive x-axis and the positive y-axis.

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3. Find the linearization (linear approximation) of the vector-valued function

f(x, y) =

 ex�y

exy

�

at (2, 1).

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4. Consider the following function defined in all of R2. Find all candidates for local extrema and use the second order test to determine the type (maximum, minimum,

saddle point or unknown).

f(x, y) = xy2 � 4y2 � 4x.

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5. Find all global maxima and global minima of

f(x, y) = x2 � xy � y2 + x + y

on the set

D = {(x, y) : 0  x  1, 0  y  1}.

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