Exam 7: Functions of Several Variables

Determine the maximum value of f(x, y) = 4 - $x ^ { 2 }$ - $y ^ { 2 }$ subject to the constraint $y = 3 x - 4$ . Enter your answer as exactly just a, b where a is the maximum and b is the Lagrange multiplier as either integers or reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ (no words or labels).

Let f(x, y) = ln(y + 2) + $e ^ { 2 x y }$ . Compute f(0, -1).

A tennis racket manufacturer produces two types of rackets, standard and competition. The weekly revenue function, in dollars, for x standard rackets and y competition rackets is given by R(x, y) = 54x + 2xy + 398y - 2 $x ^ { 2 }$ - 9 $y ^ { 2 }$ i) How many of each type of racket must be produced each week to maximize revenue? Ii) What is the maximum weekly revenue?

What is the least squares error E for the points (1, -3), (-2, 5), and (0, 10) and the line $y = A x + B ?$

A certain manufacturer can produce f(x, y) = 10(6 $x ^ { 3 }$ + $y ^ { 2 }$ ) units of goods by utilizing x units of labor and y units of capital. What is the marginal productivity of labor when $x = 10$ and $\mathrm { y } = 20 ?$

Suppose that a company can produce P(x, y) = 50 $\sqrt { \frac { \left( x ^ { 3 } + y ^ { 3 } \right) } { 10 } }$ items using x units of labor and y units of capital. What is the productivity of capital when x = 10 and $y = 20$ ?

Find the pairs (x, y) that give the extreme values of 2x + 10y, subject to the constraint $4 x ^ { 2 } + 5 y ^ { 2 } = 8400$ using the method of Lagrange multipliers. Enter your answer as exactly just (a, b), (c, d) where a > c (no words).

Let f(x, y) = $4 x ^ { 2 }$ + 2 $y ^ { 2 }$ + 3xy. Find $\frac { \partial f } { \partial y }$ . Enter just a polynomial in y plus or minus a polynomial in x both in standard form (no label, no parentheses).

Let f(x, y) = $x ^ { 2 }$ + y. Find $\frac { \partial f } { \partial y }$ . Enter just an integer.

Let H(x, y) = $\frac { 3 x y } { x ^ { 2 } - y }$ . Find $\frac { \partial ^ { 2 } \mathrm { H } } { \partial \mathrm { y } ^ { 2 } }$ .

Calculate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x ^ { 2 } } ^ { x } ( x - 1 ) d y d x$

Let f(x, y, z) = ln( $x y ^ { 2 }$ $z ^ { 3 }$ ). Find $\frac { \partial f } { \partial z }$ . Enter your answer in the unlabeled form $a z$

Calculate the iterated integral $\int _ { 0 } ^ { 2 } \left( \int _ { 0 } ^ { x } ( x + 2 y ) d y \right)$ dx. Enter just a reduced fraction of form $\frac { a } { b }$ .

Let Q(x, y) = $x ^ { 2 }$ y + $y ^ { 3 }$ $x ^ { 4 }$ . The point (1, 0) is

Calculate the iterated integral $\int _ { 0 } ^ { 1 } \left( \int _ { 0 } ^ { 1 } \left( x ^ { 3 } + y ^ { 2 } + x y \right) d y \right)$ dx. Enter just a reduced fraction of form $\frac { a } { b }$ .

Let f(x, y, z) = xyz. Find the point(s) where f(x, y, z) may have a possible relative maximum or minimum, subject to the constraint x + 6y + 3z = 36 and where x > 0, y > 0, z > 0. Use the method of Lagrange multipliers. Enter your answer as just (a, b, c) where a, b, c are all integers.

Let f(x, y) = $x ^ { 2 }$ + 2xy + $e ^ { y }$ . Find $\frac { \partial f } { \partial y }$ . Enter just a polynomial in x plus or minus a polynomial in $e ^ { y }$ both in standard form (do not label, no parentheses).

Let f(x, y) = ln(x + 2y). Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ . Is $- ( x + 2 y ) ^ { 2 }$ the correct answer?

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ - 2 $y ^ { 2 }$ + 4x - 6y + 8 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are either integers or reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Let f(x, y) = $( x + y ) ^ { 2 }$ - $( x + y ) ^ { 3 }$ . Find $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ . Is $- 6 ( x + y ) + 2$ the correct answer?