Exam 7: Functions of Several Variables

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4). Enter just a reduced fraction $\frac { a } { b }$ .

A rectangular garden is to be surrounded on three sides by a fence costing $5 per foot and on one side by a stone wall costing $15 per foot. Let x be the length of the side with the stone wall, and let y be the length of each of the other three sides. Express the cost of enclosing the garden as a function of two variables.

Use the method of your choice to obtain the formula for the least-squares line to fit the data (0, 6), (1, 3), $( 2,3 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Maximize the function f(x, y) = $x ^ { 2 }$ - $y ^ { 2 }$ subject to the constraint $y - x ^ { 2 } = - \frac { 1 } { 2 } , x , y > 0$ Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Minimize the function f(x) = x + y, subject to the constraint xy = 100, x > 0, y > 0. Use the method of Lagrange multipliers. Enter your answer exactly as just (a, b), c where (a, b) gives the minimum and c is the Lagrange multiplier as a reduced fraction of form $\frac { d } { e }$ (no words or labels).

Let f(x, y) = 5 $x ^ { 2 }$ - 5 $y ^ { 2 }$ + 2xy + 34x + 38y + 12. At which point does f(x, y) have a possible maximum or minimum value?

Design a cylindrical can of volume 100 cubic units that requires a minimum amount of aluminum; that is, the can is to have a minimum surface area. Enter your answer exactly as just r, h where r is exactly of form $\sqrt [ 3 ] { \frac { a } { b } }$ representing radius, and h is exactly of form $c \sqrt [ 3 ] { \frac { \mathrm { d } } { \mathrm { e } } }$ representing height (no labels, words, or units).

Let f(x, y) = $\frac { x } { y + 1 }$ . Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ . Enter just ± $( \mathrm { P } ( \mathrm { y } ) ) ^ { \mathrm { a } }$ where P is a polynomial in standard form (do not label).

Let f(x, y) = $\frac { \ln x y } { y }$ . Find $\frac { \partial f } { \partial x }$ . Is $( x y ) ^ { - 1 }$ the correct answer?

Let f(x, y,) = xy + 5. Compute f(1, 2 + k) - f(1, 2). Enter a polynomial in k in standard form.

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

Are these the level curves of heights 0, 1, 2, and 3 for $f ( x , y ) = 3 x - 5 y + 1$ ?

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { 0 } ^ { 1 / x ^ { 2 } } x ^ { 3 } e ^ { x ^ { 3 } y } d y \right)$ dx. Enter your answer exactly in the form $e ^ { a }$ ± b - $\mathrm { e } ^ { \mathrm { C } }$ .

Let E = $( 2 A + B - 3 ) ^ { 2 }$ + (A + B + 2) + (4A + B - 1 $)^2$ . What is $\frac { \partial \mathrm { E } } { \partial \mathrm { A } }$ ?

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Let f(x, y) = x $y ^ { 2 }$ + $x ^ { 2 }$ . Simplify $\frac { f ( x + h , y ) - f ( x , y ) } { h }$ for h ≠ 0. Enter your answer exactly in the form: $a x + y ^ { n } + b$

Maximize f(x, y, z) = x + y + z subject to the constraint $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 , x , y , z > 0$ Enter your answer as just $3 ^a$ where a is a reduced fraction of form $\frac { b } { c }$ .

Let R be the rectangle consisting of all points (x, y) such that 1 ≤ x ≤ 25, 1 ≤ y ≤ 16. Calculate $\iint _ { R } 5 \sqrt { x y } d y d x$