Exam 7: Calculus of Several Variables

Let f (x, y) = x ln(1 + 2x - 5y). Find $f _ { x x } ( x , y )$ .

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 6 x + y ^ { 2 } - 8 y$ .

If $z = x ^ { 2 } + y ^ { 2 }$ , $x = 5 \sqrt { t }$ , and $y = t ^ { 2 } - t$ , find $\frac { d z } { d t }$ when t = 9.

Compute $f _ { y }$ for $f ( x , y ) = 6 x e ^ { y }$ .

$f ( x , y ) = 4 x ^ { 2 } - y$ . Compute f (3, 0).

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1)

Compute f (7, 1) if $f ( x , y ) = 7 x y ^ { 3 }$

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = 8 x ^ { 2 } - 4 y ^ { 2 }$ .

Find the minimum value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 4 y + 4$ on the hyperbola $x ^ { 2 } - y ^ { 2 } = 4$ .

Compute f (2, 0) if $f ( x , y ) = \frac { x } { e ^ { y } }$

What is the domain of $f ( x , y ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } - 16 \right) ^ { 1 / 2 } }$ ?

Compute f (6, 0) if $f ( x , y ) = \frac { x } { e ^ { y } }$ .

At a certain factory, the output Q is related to inputs x and y by the expression $Q ( x , y ) = 4 x ^ { 4 } + 8 x ^ { 2 } y ^ { 4 } + 7 y ^ { 4 }$ If 0 $\le$ x $\le$ 7 and 0 $\le$ y $\le$ 8, what is the average output of the factory? (Approximate to 2 decimal places.)

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 4 } + y ^ { 4 } - 2 x ^ { 2 } - 2 y ^ { 2 }$ .

Compute f (ln 3, ln 6) if $f ( x , y ) = e ^ { 2 x + y }$

Compute $f _ { x }$ for $f ( x , y ) = 5 x y ^ { 3 }$

Compute f_x for f (x, y) = 3x⁸y - 4x +e^xy.

Let $f ( x , y ) = y e ^ { - 5 x ^ { 2 } y }$ . Find $f _ { y y } ( x , y )$ .

What is the domain of $f ( x , y ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } - 64 \right) ^ { 1 / 2 } }$ ?