Exam 7: Functions of Several Variables

Maximize the function f(x, y) = 2x + y subject to the constraint $x ^ { 2 } + y ^ { 2 } = 1$ Enter your answer exactly in the form $\left( a \sqrt { \frac { b } { c } } , \sqrt { \frac { d } { e } } \right)$ $\left( - h \sqrt { \frac { i } { j } } , \sqrt { \frac { k } { l } } \right)$

Let f(x, y, z) = $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ . Find $\frac { \partial f } { \partial z }$ . Enter your answer as a polynomial in $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ in standard form (unlabeled).

Let f(x, y) = $e ^ { x ^ { 2 } y }$ . Find $\frac { \partial f } { \partial y }$ . Enter your answer exactly as just $e ^ { x ^ { 2 } y }$ (P(x)) where P is a polynomial in x in standard form (do not label).

Suppose the number of leather bags produced by a certain firm, utilizing x units of raw materials and y units of labor, is given by the formula f(x, y) = 10 $x ^ { 1 / 2 }$ $y ^ { 1 / 2 }$ . What happens to the production of bags if supplies of both raw materials and labor are tripled?

Let f(x, y) = $4 x ^ { 2 }$ + 2 $y ^ { 2 / 3 }$ + $e ^ { x ^ { 4 / 3 } }$ . Find $\frac { \partial f } { \partial y }$ . Enter just a power function in y in standard form (do not label).

Let f(x, y) = xy. Find $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ . Is $y , x$ correct in the corresponding order?

Let F(x, y, z) = $\frac { x z } { y ^ { 2 } z + x }$ - $5 x ^ { 2 }$ $y ^ { 3 }$ . Compute $\frac { \partial \mathrm { F } } { \partial \mathrm { y } }$ (-1, 0, 1).

Determine the minimum of f(x, y) = $x ^ { 2 }$ + 2 $y ^ { 2 }$ subject to the constraint x - 2y + 3 = 0. Enter your answer exactly as just a, b where a is the minimum and b is the Lagrange multiplier, both as integers (no labels).

$\text { Let } f ( x , y ) = x ^ { 2 } y + y ^ { 2 } x + 2 x y \text {. Find }$ Enter just a polynomial in x plus or minus a polynomial in y plus or minus two, both polynomials in standard form (no label, no parentheses).

Let f(x, y) = $\frac { 1 } { 2 }$ $x ^ { 2 }$ ∙ $e ^ { x / y }$ . Find $\frac { \partial f } { \partial y }$ .

Let f(x, y) = $x ^ { 4 }$ - $y ^ { 2 }$ + $3 x ^ { 2 }$ + 2y - 7. The first partial derivatives of f(x, y) are zero at the points $( 0,1 )$ and $( - 1,1 )$ Use the second derivative test to determine the nature of f(x, y) at each of these points.

Let f(x, y) = $\frac { x } { y }$ - 2xy. Compute f $\left( 1 + h , \frac { 1 } { 2 } \right)$ . Enter your answer as a polynomial in h in standard form.

Find the formula that gives the least squares error for the points (-2, -3), (1, -1), (3, 4).

The function H(x, y) = $x ^ { 4 }$ - 9 $y ^ { 2 }$ - 2 $x ^ { 2 }$ y + 20y + 4 has

An artist produces two items for sale. Each unit of item I costs $50 to produce, while each unit of item II costs $200. The revenue function is $R ( x , y ) = 40 x + 7 x y + 80 y ^ { 2 } + 10 y$ , where x is units of item I and y is units of item II. Suppose the artist has only $1000 to spend on production. Which of the following is the Lagrange function the artist should use to determine what combination of production amounts (x, y) will yield maximum profits subject to the constraint that his costs must equal $1000.

Let f(x, y) = y(x + $e y - x$ ). Compute $\frac { \partial f } { \partial x }$ (2, 3). Enter your answer exactly as just a(b ± e).

Let f(x, y, z) = $e ^ { x y ^ { 2 } } z ^ { 3 }$ . Find $\frac { \partial f } { \partial z }$ . Is $3 x y ^ { 2 } z ^ { 2 } e ^ { x y ^ { 2 } } z ^ { 3 }$ the correct answer?

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute f(3, 4). Enter just an integer.

$\text { Let } f ( x , y ) = x ^ { 2 } y - x ^ { 2 }$ .Is $y ^ { 2 } + 2 x y$ the correct answer?

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 } \frac { 1 } { x y + y } d x \right)$ dy. Enter in the form $( \ln a ) ^ { b }$ .