Question 1

Let g(x, t, z) = $\frac { 1 } { t }$ (x - $z ^ { 2 }$ t). Find $\frac { \partial ^ { 2 } \mathrm {~g} } { \partial \mathrm { z } \partial \mathrm { t } }$ .

Accepted Answer

A)  -2zt 
B)  0 
C)  - $\frac { 2 z } { t }$ + 1 
D)  -2zt + $t ^ { 2 }$ 
E)  none of these 
A)  -2zt 
B)  0 
C)  - $\frac { 2 z } { t }$ + 1 
D)  -2zt + $t ^ { 2 }$ 
E)  none of these

Question 2

The table below gives the height and weight of four randomly selected University undergraduate women. What is the least squares error E for these data points? $\begin{array} { r r r r r } 
\text { height } ( \mathrm { cm } ) & 150 & 155 & 165 & 170 \
\hline \text { weight } ( \mathrm { kg } ) & 60 & 55 & 70 & 62
\end{array}$

Accepted Answer

A)  E = (150A + 60B $)^2$ + (155A + 55B $)^2$ + (165A + 70B $)^2$ + (170A + 62B $)^2$ 
B)  E = 150(A + B - 60 $)^2$ + 155(A + B - 55 $)^2$ + 165(A + B - 70 $)^2$ + 170(A + B - 62 $)^2$ 
C)  E = (150A - 60B $)^2$ + (155A - 55B $)^2$ + (165A - 70B $)^2$ + (170A - 62B $)^2$ 
D)  E = (150A + B - 60 $)^2$ + (155A + B - 55 $)^2$ + (165A + B - 70 $)^2$ + (170A + B - 62 $)^2$ 
E)  none of these 
A)  E = (150A + 60B $)^2$ + (155A + 55B $)^2$ + (165A + 70B $)^2$ + (170A + 62B $)^2$ 
B)  E = 150(A + B - 60 $)^2$ + 155(A + B - 55 $)^2$ + 165(A + B - 70 $)^2$ + 170(A + B - 62 $)^2$ 
C)  E = (150A - 60B $)^2$ + (155A - 55B $)^2$ + (165A - 70B $)^2$ + (170A - 62B $)^2$ 
D)  E = (150A + B - 60 $)^2$ + (155A + B - 55 $)^2$ + (165A + B - 70 $)^2$ + (170A + B - 62 $)^2$ 
E)  none of these

Question 3

Use partial derivatives to obtain the formula for the best least-squares fit to the data points (0, 3), (2, 5), $( 4,5 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Accepted Answer

The answer of Use partial derivatives to obtain the formula...

Question 4

Let R be the rectangle consisting of all points (x, y) such that 2 ≤ x ≤ 3, 0 ≤ y ≤ 2. 
Calculate $\iint _ { R } ( x + y ) d y d x$ Enter just an integer.

Accepted Answer

The answer of Let R be the rectangle consisting of...

Question 5

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 2 } ^ { 4 } \frac { x } { y } d y d x$

Accepted Answer

A)  6 ln 2 
B)  $\frac { 3 } { 2 }$ ln 2 
C)  $\frac { 3 } { 2 }$ ln 4 
D)  $\frac { 1 } { 2 }$ ln 2 
E)  none of these 
A)  6 ln 2 
B)  $\frac { 3 } { 2 }$ ln 2 
C)  $\frac { 3 } { 2 }$ ln 4 
D)  $\frac { 1 } { 2 }$ ln 2 
E)  none of these

Question 6

Let G(x, r, t) = 2 $x ^ { 2 }$ t + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\sqrt { x t }$ . Find $\frac { \partial G } { \partial x }$ .

Accepted Answer

A)  4xt - $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $\frac { 3 \mathrm { r } ^ { 2 } \mathrm { t } } { 2 \sqrt { x t } }$ 
B)  4x - $\frac { \mathrm { r } ^ { 3 } } { 2 \sqrt { \mathrm { xt } } }$ 
C)  $4 x t - \frac { r ^ { 3 } t } { 2 \sqrt { x t } }$ 
D)  4xt + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\frac { 1 } { 2 \sqrt { x t } }$ 
E)  none of these 
A)  4xt - $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $\frac { 3 \mathrm { r } ^ { 2 } \mathrm { t } } { 2 \sqrt { x t } }$ 
B)  4x - $\frac { \mathrm { r } ^ { 3 } } { 2 \sqrt { \mathrm { xt } } }$ 
C)  $4 x t - \frac { r ^ { 3 } t } { 2 \sqrt { x t } }$ 
D)  4xt + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\frac { 1 } { 2 \sqrt { x t } }$ 
E)  none of these

Question 7

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0.
Enter your answer as a polynomial in k in standard form.

Accepted Answer

The answer of \(\text { Let } f ( x...

Question 8

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

Accepted Answer

The answer of Use the method of your choice to...

Question 9

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { - 2 } ^ { 1 } 32 x ^ { 3 } y ^ { 3 } d y \right)$ dx.

Accepted Answer

A)  -450 
B)  30 
C)  450 
D)  -480 
A)  -450 
B)  30 
C)  450 
D)  -480

Question 10

A petroleum company has a Cobb-Douglas production function $f ( x , y ) = 70 x ^ { 2 / 5 } y ^ { 3 / 5 }$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of petroleum produced when 1200 units of labor and 2100 units of capital are used.

Accepted Answer

A)  175,174 units 
B)  150,000 units 
C)  105,074 units 
D)  117,517 units 
A)  175,174 units 
B)  150,000 units 
C)  105,074 units 
D)  117,517 units

Question 11

Let f(x, y) = $y e ^ { x }$ + x $y ^ { 2 }$ . At which point does f(x, y) have a possible maximum or minimum value?

Accepted Answer

A)  $\left( \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } \mathrm { e } \sqrt { 2 } / 2 } { 2 } \right)$ 
B)  (0, 0) 
C)  $\left( \frac { 1 } { 2 } , - \frac { \mathrm { e } ^ { 1 / 2 } } { 2 } \right)$ 
D)  $\left( \frac { 1 } { 2 } , - \mathrm { e } ^ { 1 / 2 } \right)$ 
E)  none of these 
A)  $\left( \frac { \sqrt { 2 } } { 2 } , - \frac { \sqrt { 2 } \mathrm { e } \sqrt { 2 } / 2 } { 2 } \right)$ 
B)  (0, 0) 
C)  $\left( \frac { 1 } { 2 } , - \frac { \mathrm { e } ^ { 1 / 2 } } { 2 } \right)$ 
D)  $\left( \frac { 1 } { 2 } , - \mathrm { e } ^ { 1 / 2 } \right)$ 
E)  none of these

Question 12

Maximize the function f(x, y) = $e ^ { x y }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = 18 , x , y > 0$ Enter your answer as just $e ^ { a }$ .

Accepted Answer

The answer of Maximize the function f(x, y) = \(e...

Question 13

Let f(x, y) = $x ^ { 3 }$ y + $e ^ { x + 3 y }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ (1, 0).

Accepted Answer

A)  6 
B)  3 + 3e 
C)  6 + e 
D)  3 + e 
E)  none of these 
A)  6 
B)  3 + 3e 
C)  6 + e 
D)  3 + e 
E)  none of these

Question 14

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

Accepted Answer

The answer of Let f(x, y) = x \(e ^...

Question 15

Let f(x, y, z) = $y ^ { 3 }$ z - $x ^ { 2 }$ y + $\sqrt { x y z }$ - $\frac { 1 } { x }$ . Find $\frac { \partial f } { \partial y }$ .

Accepted Answer

A)  $3 y ^ { 2 }$ - 2x + 1 
B)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\frac { x z } { 2 \sqrt { x y z } }$ 
C)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\sqrt { x _ { 7 } }$ 
D)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\frac { x z } { \sqrt { x y z } }$ + $\frac { 1 } { x ^ { 2 } }$ 
E)  none of these 
A)  $3 y ^ { 2 }$ - 2x + 1 
B)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\frac { x z } { 2 \sqrt { x y z } }$ 
C)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\sqrt { x _ { 7 } }$ 
D)  $3 y ^ { 2 }$ z - $x ^ { 2 }$ + $\frac { x z } { \sqrt { x y z } }$ + $\frac { 1 } { x ^ { 2 } }$ 
E)  none of these

Question 16

Assume that a manufacturer has productivity function P(l, c) where l and c are the amounts of labor and capital utilized. Which of the following indicates that a slight increase in the amount of labor utilized will result in an increase in productivity of 3 units.

Accepted Answer

A)  $\frac { \partial \mathrm { P } } { \partial \mathrm { l } }$ (100, 75) = 3 
B)  P(10, 75) = 3 
C)  P(1, 75) = 3 
D)  $\frac { \partial P } { \partial c }$ (100, 75) = 3 
E)  none of these 
A)  $\frac { \partial \mathrm { P } } { \partial \mathrm { l } }$ (100, 75) = 3 
B)  P(10, 75) = 3 
C)  P(1, 75) = 3 
D)  $\frac { \partial P } { \partial c }$ (100, 75) = 3 
E)  none of these

Question 17

Find all points (x, y) where f(x, y) = 3xy - $x ^ { 2 }$ - $y ^ { 2 }$ - 2x - y + 3 has a possible relative maximum or minimum.
Enter your answer as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Accepted Answer

The answer of Find all points (x, y) where f(x,...

Question 18

Calculate the iterated integral $\int _ { 0 } ^ { \ln 2 } \left( \int _ { 1 } ^ { 2 } x y + y e ^ { x y } d x \right)$ dy.

Accepted Answer

A)  $\frac { 3 } { 2 }$ ln - $e ^ { 2 } \ln 2$ - $\frac { 3 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ ln 2 - $\frac { 1 } { 2 }$ $\mathrm{ex}^{2}$  - $\frac { 5 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ (ln 2) + $\frac { 1 } { 2 }$ 
D)  $\frac { 3 } { 4 }$ $( \ln 2 ) ^ { 2 }$ + $\frac { 1 } { 2 }$ 
E)  none of these 
A)  $\frac { 3 } { 2 }$ ln - $e ^ { 2 } \ln 2$ - $\frac { 3 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ ln 2 - $\frac { 1 } { 2 }$ $\mathrm{ex}^{2}$  - $\frac { 5 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ (ln 2) + $\frac { 1 } { 2 }$ 
D)  $\frac { 3 } { 4 }$ $( \ln 2 ) ^ { 2 }$ + $\frac { 1 } { 2 }$ 
E)  none of these

Question 19

Suppose that a retailer sells f(p, a) units of an item, where p is the price per unit of the item and a is the amount of money spent on advertising that item. Which of the following indicates that as the amount spent on advertising is decreased, demand for the item also decreases?

Accepted Answer

A)  $\frac { \partial f } { \partial a }$ (50, 75) = -10 
B)  $\frac { \partial f } { \partial p }$ (50, 25) = -1 
C)  $\frac { \partial f } { \partial p }$ (50, -1) = -5 
D)  $\frac { \partial f } { \partial a }$ (50, 25) = 10 
A)  $\frac { \partial f } { \partial a }$ (50, 75) = -10 
B)  $\frac { \partial f } { \partial p }$ (50, 25) = -1 
C)  $\frac { \partial f } { \partial p }$ (50, -1) = -5 
D)  $\frac { \partial f } { \partial a }$ (50, 25) = 10

Question 20

Let P(x, y, z) = xy + 2 $x ^ { 3 }$ $\sqrt { y ^ { 2 } - 1 }$ . Compute $\frac { \partial P } { \partial z }$ (2, 1, 3).

Accepted Answer

A)  2 
B)  0 
C)  3 
D)  1 
E)  none of these 
A)  2 
B)  0 
C)  3 
D)  1 
E)  none of these

Let g(x, t, z) = $\frac { 1 } { t }$ (x - $z ^ { 2 }$ t). Find $\frac { \partial ^ { 2 } \mathrm {~g} } { \partial \mathrm { z } \partial \mathrm { t } }$ .

The table below gives the height and weight of four randomly selected University undergraduate women. What is the least squares error E for these data points? height () 150 155 165 170 weight () 60 55 70 62

Use partial derivatives to obtain the formula for the best least-squares fit to the data points (0, 3), (2, 5), $( 4,5 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Let R be the rectangle consisting of all points (x, y) such that 2 ≤ x ≤ 3, 0 ≤ y ≤ 2. Calculate $\iint _ { R } ( x + y ) d y d x$ Enter just an integer.

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 2 } ^ { 4 } \frac { x } { y } d y d x$

Let G(x, r, t) = 2 $x ^ { 2 }$ t + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\sqrt { x t }$ . Find $\frac { \partial G } { \partial x }$ .

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$ Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

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

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { - 2 } ^ { 1 } 32 x ^ { 3 } y ^ { 3 } d y \right)$ dx.

A petroleum company has a Cobb-Douglas production function $f ( x , y ) = 70 x ^ { 2 / 5 } y ^ { 3 / 5 }$ where x is the utilization of labor and y is the utilization of capital. Determine the number of units of petroleum produced when 1200 units of labor and 2100 units of capital are used.

Let f(x, y) = $y e ^ { x }$ + x $y ^ { 2 }$ . At which point does f(x, y) have a possible maximum or minimum value?

Maximize the function f(x, y) = $e ^ { x y }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = 18 , x , y > 0$ Enter your answer as just $e ^ { a }$ .

Let f(x, y) = $x ^ { 3 }$ y + $e ^ { x + 3 y }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ (1, 0).

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

Let f(x, y, z) = $y ^ { 3 }$ z - $x ^ { 2 }$ y + $\sqrt { x y z }$ - $\frac { 1 } { x }$ . Find $\frac { \partial f } { \partial y }$ .

Assume that a manufacturer has productivity function P(l, c) where l and c are the amounts of labor and capital utilized. Which of the following indicates that a slight increase in the amount of labor utilized will result in an increase in productivity of 3 units.

Find all points (x, y) where f(x, y) = 3xy - $x ^ { 2 }$ - $y ^ { 2 }$ - 2x - y + 3 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Calculate the iterated integral $\int _ { 0 } ^ { \ln 2 } \left( \int _ { 1 } ^ { 2 } x y + y e ^ { x y } d x \right)$ dy.

Suppose that a retailer sells f(p, a) units of an item, where p is the price per unit of the item and a is the amount of money spent on advertising that item. Which of the following indicates that as the amount spent on advertising is decreased, demand for the item also decreases?

Let P(x, y, z) = xy + 2 $x ^ { 3 }$ $\sqrt { y ^ { 2 } - 1 }$ . Compute $\frac { \partial P } { \partial z }$ (2, 1, 3).

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 7: Functions of Several Variables

Let g(x, t, z) = 1t\frac { 1 } { t }t1​ (x - z2z ^ { 2 }z2 t). Find ∂2 g∂z∂t\frac { \partial ^ { 2 } \mathrm {~g} } { \partial \mathrm { z } \partial \mathrm { t } }∂z∂t∂2 g​ .

The table below gives the height and weight of four randomly selected University undergraduate women. What is the least squares error E for these data points? height () 150 155 165 170 weight () 60 55 70 62

Use partial derivatives to obtain the formula for the best least-squares fit to the data points (0, 3), (2, 5), (4,5).( 4,5 ) .(4,5). Enter your answer in standard point-intercept form with any fractions reduced of form ab\frac { a } { b }ba​ .

Let R be the rectangle consisting of all points (x, y) such that 2 ≤ x ≤ 3, 0 ≤ y ≤ 2. Calculate ∬R(x+y)dydx\iint _ { R } ( x + y ) d y d x∬R​(x+y)dydx Enter just an integer.

Calculate the iterated integral ∫12∫24xydydx\int _ { 1 } ^ { 2 } \int _ { 2 } ^ { 4 } \frac { x } { y } d y d x∫12​∫24​yx​dydx

Let G(x, r, t) = 2 x2x ^ { 2 }x2 t + 13\frac { 1 } { 3 }31​ t2t ^ { 2 }t2 - r3r ^ { 3 }r3 xt\sqrt { x t }xt​ . Find ∂G∂x\frac { \partial G } { \partial x }∂x∂G​ .

Let f(x,y)=x2y2−1\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1 Let f(x,y)=x2y2−1 Compute f(−1,1+k)−f(−1,1)k\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }kf(−1,1+k)−f(−1,1)​ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

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

Calculate the iterated integral ∫−12(∫−2132x3y3dy)\int _ { - 1 } ^ { 2 } \left( \int _ { - 2 } ^ { 1 } 32 x ^ { 3 } y ^ { 3 } d y \right)∫−12​(∫−21​32x3y3dy) dx.

Let f(x, y) = yexy e ^ { x }yex + x y2y ^ { 2 }y2 . At which point does f(x, y) have a possible maximum or minimum value?

Maximize the function f(x, y) = exye ^ { x y }exy subject to the constraint x2+y2=18,x,y>0x ^ { 2 } + y ^ { 2 } = 18 , x , y > 0x2+y2=18,x,y>0 Enter your answer as just eae ^ { a }ea .

Let f(x, y) = x3x ^ { 3 }x3 y + ex+3ye ^ { x + 3 y }ex+3y . Compute ∂2f∂x∂y\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }∂x∂y∂2f​ (1, 0).

Let f(x, y) = x e2ye ^ { 2 } ye2y + y e2x\mathrm { e } ^ { 2 x }e2x . Compute ∂2f∂x2\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } }∂x2∂2f​ . Enter your answer as just an unlabeled polynomial in e2x\mathrm { e } ^ { 2 x }e2x in standard form.

Let f(x, y, z) = y3y ^ { 3 }y3 z - x2x ^ { 2 }x2 y + xyz\sqrt { x y z }xyz​ - 1x\frac { 1 } { x }x1​ . Find ∂f∂y\frac { \partial f } { \partial y }∂y∂f​ .

Assume that a manufacturer has productivity function P(l, c) where l and c are the amounts of labor and capital utilized. Which of the following indicates that a slight increase in the amount of labor utilized will result in an increase in productivity of 3 units.

Find all points (x, y) where f(x, y) = 3xy - x2x ^ { 2 }x2 - y2y ^ { 2 }y2 - 2x - y + 3 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are reduced fractions of form cd\frac { \mathrm { c } } { \mathrm { d } }dc​ .

Calculate the iterated integral ∫0ln⁡2(∫12xy+yexydx)\int _ { 0 } ^ { \ln 2 } \left( \int _ { 1 } ^ { 2 } x y + y e ^ { x y } d x \right)∫0ln2​(∫12​xy+yexydx) dy.

Suppose that a retailer sells f(p, a) units of an item, where p is the price per unit of the item and a is the amount of money spent on advertising that item. Which of the following indicates that as the amount spent on advertising is decreased, demand for the item also decreases?

Let P(x, y, z) = xy + 2 x3x ^ { 3 }x3 y2−1\sqrt { y ^ { 2 } - 1 }y2−1​ . Compute ∂P∂z\frac { \partial P } { \partial z }∂z∂P​ (2, 1, 3).

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

The Trigonometric Functions

Techniques of Integration

Differential Equations

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Let g(x, t, z) = $\frac { 1 } { t }$ (x - $z ^ { 2 }$ t). Find $\frac { \partial ^ { 2 } \mathrm {~g} } { \partial \mathrm { z } \partial \mathrm { t } }$ .

Use partial derivatives to obtain the formula for the best least-squares fit to the data points (0, 3), (2, 5), $( 4,5 ) .$ Enter your answer in standard point-intercept form with any fractions reduced of form $\frac { a } { b }$ .

Let R be the rectangle consisting of all points (x, y) such that 2 ≤ x ≤ 3, 0 ≤ y ≤ 2. Calculate $\iint _ { R } ( x + y ) d y d x$ Enter just an integer.

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 2 } ^ { 4 } \frac { x } { y } d y d x$

Let G(x, r, t) = 2 $x ^ { 2 }$ t + $\frac { 1 } { 3 }$ $t ^ { 2 }$ - $r ^ { 3 }$ $\sqrt { x t }$ . Find $\frac { \partial G } { \partial x }$ .

$\text { Let } f ( x , y ) = x ^ { 2 } y ^ { 2 } - 1$ Compute $\frac { f ( - 1,1 + k ) - f ( - 1,1 ) } { k }$ for k ≠ 0. Enter your answer as a polynomial in k in standard form.

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

Calculate the iterated integral $\int _ { - 1 } ^ { 2 } \left( \int _ { - 2 } ^ { 1 } 32 x ^ { 3 } y ^ { 3 } d y \right)$ dx.

Let f(x, y) = $y e ^ { x }$ + x $y ^ { 2 }$ . At which point does f(x, y) have a possible maximum or minimum value?

Maximize the function f(x, y) = $e ^ { x y }$ subject to the constraint $x ^ { 2 } + y ^ { 2 } = 18 , x , y > 0$ Enter your answer as just $e ^ { a }$ .

Let f(x, y) = $x ^ { 3 }$ y + $e ^ { x + 3 y }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } }$ (1, 0).

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

Let f(x, y, z) = $y ^ { 3 }$ z - $x ^ { 2 }$ y + $\sqrt { x y z }$ - $\frac { 1 } { x }$ . Find $\frac { \partial f } { \partial y }$ .

Find all points (x, y) where f(x, y) = 3xy - $x ^ { 2 }$ - $y ^ { 2 }$ - 2x - y + 3 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Calculate the iterated integral $\int _ { 0 } ^ { \ln 2 } \left( \int _ { 1 } ^ { 2 } x y + y e ^ { x y } d x \right)$ dy.

Let P(x, y, z) = xy + 2 $x ^ { 3 }$ $\sqrt { y ^ { 2 } - 1 }$ . Compute $\frac { \partial P } { \partial z }$ (2, 1, 3).