Question 1

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .

Accepted Answer

A)  $2$ 
B)  $11$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 13 } { 15 }$ 
E)  $\frac { 11 } { 30 }$ 
A)  $2$ 
B)  $11$ 
C)  $\frac { 13 } { 2 }$ 
D)  $\frac { 13 } { 15 }$ 
E)  $\frac { 11 } { 30 }$

Question 2

Evaluate $f _ { x }$ and $f _ { y }$ for the function $f ( x , y ) = 7 x ^ { 3 } + x y - y ^ { 3 }$ at the point $( 4 , - 6 )$ .

Accepted Answer

A)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -132 
B)  $f _ { x } ( 4 , - 6 ) =$ 360 and $f _ { y } ( 4 , - 6 ) =$ -104 
C)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -104 
D)  $f _ { x } ( 4 , - 6 ) =$ 546 and $f _ { y } ( 4 , - 6 ) =$ 344 
E)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ 112 
A)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -132 
B)  $f _ { x } ( 4 , - 6 ) =$ 360 and $f _ { y } ( 4 , - 6 ) =$ -104 
C)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ -104 
D)  $f _ { x } ( 4 , - 6 ) =$ 546 and $f _ { y } ( 4 , - 6 ) =$ 344 
E)  $f _ { x } ( 4 , - 6 ) =$ 330 and $f _ { y } ( 4 , - 6 ) =$ 112

Question 3

Test for relative extrema and saddle points. $z = x ^ { 2 } - 8 x y + y ^ { 2 } + 120 x$

Accepted Answer

A) saddle point at $( - 4 , - 16 , - 720 )$ 
B) saddle point at $( 4,16,240 )$ 
C) saddle point at $( 0,0,0 )$ 
D) relative minimum at $( 4 , - 16,1264 )$ 
E) relative minimum at $( 16,128,2176 )$ 
A) saddle point at $( - 4 , - 16 , - 720 )$ 
B) saddle point at $( 4,16,240 )$ 
C) saddle point at $( 0,0,0 )$ 
D) relative minimum at $( 4 , - 16,1264 )$ 
E) relative minimum at $( 16,128,2176 )$

Question 4

Examine the function $f ( x , y ) = 5 x ^ { 2 } - 8 y ^ { 2 } - 3 x + 8 y + 4$ for relative extrema.

Accepted Answer

A) saddle point at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
B) relative minimum at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
C) saddle point at $\left( - \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
D) relative maximum at $\left( \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$ 
E) saddle point at $\left( - \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$ 
A) saddle point at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
B) relative minimum at $\left( \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
C) saddle point at $\left( - \frac { 3 } { 10 } , \frac { 1 } { 2 } \right)$ 
D) relative maximum at $\left( \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$ 
E) saddle point at $\left( - \frac { 3 } { 10 } , - \frac { 1 } { 2 } \right)$

Question 5

An agronomist used four test plots to determine the relationship between the wheat yield $y$ (in bushels per acre) and the amount of fertilizer $x$ (in pounds per acre). The results are shown in the table. $\begin{array} { | c | c | c | c | c | } 
\hline \text { Fertilizer, } x & 100 & 150 & 200 & 250 \
\hline \text { Yield, } y & 35 & 44 & 50 & 56 \
\hline
\end{array}$ (a) Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the data. (b) Estimate the yield for a fertilizer application of 160 pounds per acre.

Accepted Answer

A)  $y = 0.15 x + 15.1$ 
B)  $y = 0.155 x + 21.1$ 
C)  $y = 0.138 x + 22.1$ 
D)  $y = 0.052 x + 34$ 
E)  $y = 0.234 x + 17.5$ 
A)  $y = 0.15 x + 15.1$ 
B)  $y = 0.155 x + 21.1$ 
C)  $y = 0.138 x + 22.1$ 
D)  $y = 0.052 x + 34$ 
E)  $y = 0.234 x + 17.5$

Question 6

Find the critical points of the function $f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Accepted Answer

A) relative minimum at $( - 4 , - 7 , - 2 )$ 
B) relative maximum at $( - 4 , - 7 , - 2 )$ 
C) relative minimum at $( 4,7,2 )$ 
D) relative maximum at $( 4,7,2 )$ 
E) no relative extrema 
A) relative minimum at $( - 4 , - 7 , - 2 )$ 
B) relative maximum at $( - 4 , - 7 , - 2 )$ 
C) relative minimum at $( 4,7,2 )$ 
D) relative maximum at $( 4,7,2 )$ 
E) no relative extrema

Question 7

Evaluate the function at the given values of the independent variables. $z = x ^ { 3 } + 2 x y + 3 y ^ { 2 } , \quad x = - 9 , y = 7$

Accepted Answer

A)  $z = - 708$ 
B)  $z = 102$ 
C)  $z = - 600$ 
D)  $z = - 568$ 
E)  $z = 174$ 
A)  $z = - 708$ 
B)  $z = 102$ 
C)  $z = - 600$ 
D)  $z = - 568$ 
E)  $z = 174$

Question 8

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .

Accepted Answer

A) 134.00 
B) 144.00 
C) 154.00 
D) 42.00 
E) 24.00 
A) 134.00 
B) 144.00 
C) 154.00 
D) 42.00 
E) 24.00

Question 9

Describe the domain and range of the function. $f ( x , y ) = \sqrt { 49 - x ^ { 2 } - y ^ { 2 } }$

Accepted Answer

A) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval (0,7) 
B) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7] 
C) domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7) 
D) domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7] 
E) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7) 
A) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval (0,7) 
B) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7] 
C) domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7) 
D) domain: The disk $x ^ { 2 } + y ^ { 2 } \leq 49$ range: The interval [0,7] 
E) domain: The disk $x ^ { 2 } + y ^ { 2 } < 49$ range: The interval [0,7)

Question 10

Sketch the region $R$ whose area is given by the following double integral. $$\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

Describe the trace of the surface given by the function below $y ^ { 2 } + z ^ { 2 } - x ^ { 2 } = 1$ in the xz-plane.

Accepted Answer

A) circle 
B) parabola 
C) line 
D) ellipse 
E) hyperbola 
A) circle 
B) parabola 
C) line 
D) ellipse 
E) hyperbola

Question 12

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .

Accepted Answer

A)  $\frac { 1 } { 180 }$ 
B)  $\frac { 1 } { 19 }$ 
C)  $\frac { 1 } { 144 }$ 
D)  $- \frac { 1 } { 72 }$ 
E)  $- \frac { 1 } { 108 }$ 
A)  $\frac { 1 } { 180 }$ 
B)  $\frac { 1 } { 19 }$ 
C)  $\frac { 1 } { 144 }$ 
D)  $- \frac { 1 } { 72 }$ 
E)  $- \frac { 1 } { 108 }$

Question 13

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. $\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A$
$R$ : trianglebounded by $y = 5 x , y = 6 x$, and $x = 5$

Accepted Answer

A)  $\frac { 2 } { 5 } ( \log ( 37 ) - \log ( 26 ) )$ 
B)  $\frac { 5 } { 2 } ( \log ( 26 ) - \log ( 37 ) )$ 
C)  $5 ( \log ( 37 ) - \log ( 26 ) )$ 
D)  $\frac { 5 } { 2 } ( \log ( 37 ) - \log ( 26 ) )$ 
E)  $\frac { 1 } { 3 } ( \log ( 37 ) - \log ( 26 ) )$ 
A)  $\frac { 2 } { 5 } ( \log ( 37 ) - \log ( 26 ) )$ 
B)  $\frac { 5 } { 2 } ( \log ( 26 ) - \log ( 37 ) )$ 
C)  $5 ( \log ( 37 ) - \log ( 26 ) )$ 
D)  $\frac { 5 } { 2 } ( \log ( 37 ) - \log ( 26 ) )$ 
E)  $\frac { 1 } { 3 } ( \log ( 37 ) - \log ( 26 ) )$

Question 14

The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 60 inches. Find the dimensions of the rectangular package of largest volume that may be sent.

Accepted Answer

A) x = 7.5, y = 7.5, z = 20 
B) x = 5, y = 5, z = 40 
C) x = 10 , y = 10 , z = 20 
D) x = 4, y = 6, z = 40 
E) x = 2, y = 12, z = 32 
A) x = 7.5, y = 7.5, z = 20 
B) x = 5, y = 5, z = 40 
C) x = 10 , y = 10 , z = 20 
D) x = 4, y = 6, z = 40 
E) x = 2, y = 12, z = 32

Question 15

A microbiologist must prepare a culture medium in which to grow a certain type of bacteria. The percent of salt contained in this medium is given by $S = 12 x y z$ where $x , y,$ and $z$ are the nutrient solutions to be mixed in the medium. For the bacteria to grow, the medium must be 13% salt. Nutrient solutions $x , y,$ and $z$ cost $1, $2, and $3 per liter, respectively. How much of each nutrient solution should be used to minimize the cost of the culture medium?

Accepted Answer

A)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \
y = \sqrt [ 3 ] { 0.065 } \approx 0.201 L \
z = \sqrt [ 3 ] { 0.065 } \approx 0.134 L
\end{array}$ 
B)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.015 } \approx 0.2466 L
\end{array}$ 
C)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.055 } \approx 0.380 L
\end{array}$ 
D)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.025 } \approx 0.292 L \
y = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \
z = \sqrt [ 3 ] { 0.115 } \approx 0.486 L
\end{array}$ 
E)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.265 } \approx 0.642 L
\end{array}$ 
A)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \
y = \sqrt [ 3 ] { 0.065 } \approx 0.201 L \
z = \sqrt [ 3 ] { 0.065 } \approx 0.134 L
\end{array}$ 
B)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.015 } \approx 0.2466 L
\end{array}$ 
C)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.065 } \approx 0.402 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.055 } \approx 0.380 L
\end{array}$ 
D)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.025 } \approx 0.292 L \
y = \sqrt [ 3 ] { 0.035 } \approx 0.327 L \
z = \sqrt [ 3 ] { 0.115 } \approx 0.486 L
\end{array}$ 
E)  $\begin{array} { l } 
x = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
y = \sqrt [ 3 ] { 0.165 } \approx 0.548 L \
z = \sqrt [ 3 ] { 0.265 } \approx 0.642 L
\end{array}$

Question 16

A firm's weekly profit (in dollars) in marketing two products is given by $P = 192 x _ { 1 } + 576 x _ { 2 } - x _ { 1 } ^ { 2 } - 5 x _ { 2 } ^ { 2 } - 2 x _ { 1 } x _ { 2 } - 5000$ where $x _ { 1 }$ and $x _ { 2 }$ represent the numbers of units of each product sold weekly. Estimate the average weekly profit when $x _ { 1 }$ varies between 40 and 50 units and $x _ { 2 }$ varies between 45 and 50 units.

Accepted Answer

A)  $\$ 12,300$ 
B)  $\$ 11,100$ 
C)  $\$ 13,400$ 
D)  $\$ 21,760$ 
E)  $\$ 16,450$ 
A)  $\$ 12,300$ 
B)  $\$ 11,100$ 
C)  $\$ 13,400$ 
D)  $\$ 21,760$ 
E)  $\$ 16,450$

Question 17

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points $( 10,2 ) , ( 1,8 ) , ( 6,9 ) , ( 4,10 ) , ( 16,4 ) , ( 17,4 )$ . Round your answer to three decimal places.

Accepted Answer

A)  $y = - 0.383 x + 9.524$ 
B)  $y = - 0.373 x + 9.514$ 
C)  $y = - 0.373 x + 9.524$ 
D)  $y = - 0.363 x + 9.524$ 
E)  $y = - 0.363 x + 9.514$ 
A)  $y = - 0.383 x + 9.524$ 
B)  $y = - 0.373 x + 9.514$ 
C)  $y = - 0.373 x + 9.524$ 
D)  $y = - 0.363 x + 9.524$ 
E)  $y = - 0.363 x + 9.514$

Question 18

The utility function $U = f ( x , y )$ is a measure of utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is $U = - 7 x ^ { 2 } + 5 x y - 2 y ^ { 2 }$ . Determine the marginal utility of product x.

Accepted Answer

A)  $5 x - 4 y$ 
B)  $- 14 x + 5 y$ 
C)  $- 14 x + 5 y - 2 y ^ { 2 }$ 
D)  $- 7 x ^ { 2 } + 5 x - 4 y$ 
E)  $- 14 x + 5 - 4 y$ 
A)  $5 x - 4 y$ 
B)  $- 14 x + 5 y$ 
C)  $- 14 x + 5 y - 2 y ^ { 2 }$ 
D)  $- 7 x ^ { 2 } + 5 x - 4 y$ 
E)  $- 14 x + 5 - 4 y$

Question 19

Evaluate the function $w = \frac { x ^ { 2 } - y z } { x y z }$ at $( - 8,6,3 )$

Accepted Answer

A)  $w = \frac { 13 } { 72 }$ 
B)  $w = - \frac { 29 } { 72 }$ 
C)  $w = - \frac { 7 } { 9 }$ 
D)  $w = - \frac { 41 } { 72 }$ 
E)  $w = - \frac { 23 } { 72 }$ 
A)  $w = \frac { 13 } { 72 }$ 
B)  $w = - \frac { 29 } { 72 }$ 
C)  $w = - \frac { 7 } { 9 }$ 
D)  $w = - \frac { 41 } { 72 }$ 
E)  $w = - \frac { 23 } { 72 }$

Question 20

Use the function $q \left( p _ { 1 } , p _ { 2 } \right) = \frac { 7 p _ { 1 } - 9 p _ { 2 } } { - 9 p _ { 1 } + 9 p _ { 2 } }$ to find $q ( - 5 , - 8 )$

Accepted Answer

A)  $q ( - 5 , - 8 ) = - \frac { 27 } { 37 }$ 
B)  $q ( - 5 , - 8 ) = \frac { 27 } { 37 }$ 
C)  $q ( - 5 , - 8 ) = - \frac { 37 } { 27 }$ 
D)  $q ( - 5 , - 8 ) = \frac { 37 } { 27 }$ 
E)  $q ( - 5 , - 8 ) = - \frac { 54 } { 37 }$ 
A)  $q ( - 5 , - 8 ) = - \frac { 27 } { 37 }$ 
B)  $q ( - 5 , - 8 ) = \frac { 27 } { 37 }$ 
C)  $q ( - 5 , - 8 ) = - \frac { 37 } { 27 }$ 
D)  $q ( - 5 , - 8 ) = \frac { 37 } { 27 }$ 
E)  $q ( - 5 , - 8 ) = - \frac { 54 } { 37 }$

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .

Evaluate $f _ { x }$ and $f _ { y }$ for the function $f ( x , y ) = 7 x ^ { 3 } + x y - y ^ { 3 }$ at the point $( 4 , - 6 )$ .

Test for relative extrema and saddle points. $z = x ^ { 2 } - 8 x y + y ^ { 2 } + 120 x$

Examine the function $f ( x , y ) = 5 x ^ { 2 } - 8 y ^ { 2 } - 3 x + 8 y + 4$ for relative extrema.

Find the critical points of the function $f ( x , y , z ) = ( x - 4 ) ^ { 2 } + ( 7 - y ) ^ { 2 } + ( z - 2 ) ^ { 2 }$ , and, from the form of the function, determine whether a relative maximum or a relative minimum occurs at each point.

Evaluate the function at the given values of the independent variables. $z = x ^ { 3 } + 2 x y + 3 y ^ { 2 } , \quad x = - 9 , y = 7$

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .

Describe the domain and range of the function. $f ( x , y ) = \sqrt { 49 - x ^ { 2 } - y ^ { 2 } }$

Sketch the region $R$ whose area is given by the following double integral. $\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$

Describe the trace of the surface given by the function below $y ^ { 2 } + z ^ { 2 } - x ^ { 2 } = 1$ in the xz-plane.

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. $\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A$ $R$ : trianglebounded by $y = 5 x , y = 6 x$ , and $x = 5$

The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 60 inches. Find the dimensions of the rectangular package of largest volume that may be sent.

Use the regression capabilities of a graphing utility or a spreadsheet to find the least squares regression line for the points $( 10,2 ) , ( 1,8 ) , ( 6,9 ) , ( 4,10 ) , ( 16,4 ) , ( 17,4 )$ . Round your answer to three decimal places.

The utility function $U = f ( x , y )$ is a measure of utility (or satisfaction) derived by a person from the consumption of two products x and y. Suppose the utility function is $U = - 7 x ^ { 2 } + 5 x y - 2 y ^ { 2 }$ . Determine the marginal utility of product x.

Evaluate the function $w = \frac { x ^ { 2 } - y z } { x y z }$ at $( - 8,6,3 )$

Use the function $q \left( p _ { 1 } , p _ { 2 } \right) = \frac { 7 p _ { 1 } - 9 p _ { 2 } } { - 9 p _ { 1 } + 9 p _ { 2 } }$ to find $q ( - 5 , - 8 )$

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Exam 14: Functions of Several Variables

Use a double integral to find the area of the region bounded by the graphs of y=x13/2y = x ^ { 13 / 2 }y=x13/2 and y=xy = xy=x .

Evaluate fxf _ { x }fx​ and fyf _ { y }fy​ for the function f(x,y)=7x3+xy−y3f ( x , y ) = 7 x ^ { 3 } + x y - y ^ { 3 }f(x,y)=7x3+xy−y3 at the point (4,−6)( 4 , - 6 )(4,−6) .

Test for relative extrema and saddle points. z=x2−8xy+y2+120xz = x ^ { 2 } - 8 x y + y ^ { 2 } + 120 xz=x2−8xy+y2+120x

Examine the function f(x,y)=5x2−8y2−3x+8y+4f ( x , y ) = 5 x ^ { 2 } - 8 y ^ { 2 } - 3 x + 8 y + 4f(x,y)=5x2−8y2−3x+8y+4 for relative extrema.

Evaluate the function at the given values of the independent variables. z=x3+2xy+3y2,x=−9,y=7z = x ^ { 3 } + 2 x y + 3 y ^ { 2 } , \quad x = - 9 , y = 7z=x3+2xy+3y2,x=−9,y=7

Evaluate the double integral ∫04∫03(3x+4y)dydx\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x∫04​∫03​(3x+4y)dydx .

Describe the domain and range of the function. f(x,y)=49−x2−y2f ( x , y ) = \sqrt { 49 - x ^ { 2 } - y ^ { 2 } }f(x,y)=49−x2−y2​

Sketch the region RRR whose area is given by the following double integral. ∫04∫x2dydx\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x∫04​∫x​2​dydx

Describe the trace of the surface given by the function below y2+z2−x2=1y ^ { 2 } + z ^ { 2 } - x ^ { 2 } = 1y2+z2−x2=1 in the xz-plane.

Evaluate the double integral ∫0∞∫0∞xye−(4x2+9y2)dxdy\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y∫0∞​∫0∞​xye−(4x2+9y2)dxdy .

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. ∬Ryx2+y2dA\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A∬R​x2+y2y​dA RRR : trianglebounded by y=5x,y=6xy = 5 x , y = 6 xy=5x,y=6x , and x=5x = 5x=5

The sum of the length (denote by z) and the girth (perimeter of a cross section) of packages carried by a delivery service cannot exceed 60 inches. Find the dimensions of the rectangular package of largest volume that may be sent.

Evaluate the function w=x2−yzxyzw = \frac { x ^ { 2 } - y z } { x y z }w=xyzx2−yz​ at (−8,6,3)( - 8,6,3 )(−8,6,3)

Use the function q(p1,p2)=7p1−9p2−9p1+9p2q \left( p _ { 1 } , p _ { 2 } \right) = \frac { 7 p _ { 1 } - 9 p _ { 2 } } { - 9 p _ { 1 } + 9 p _ { 2 } }q(p1​,p2​)=−9p1​+9p2​7p1​−9p2​​ to find q(−5,−8)q ( - 5 , - 8 )q(−5,−8)

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

Applications of the Derivative

Further Applications of the Derivative

Derivatives of Exponential and Logarithmic Functions

Integration and Its Applications

Techniques of Integration

Trigonometric Functions Web

Series and Taylor Polynomials Web

Probability Web

Filters

Use a double integral to find the area of the region bounded by the graphs of $y = x ^ { 13 / 2 }$ and $y = x$ .

Evaluate $f _ { x }$ and $f _ { y }$ for the function $f ( x , y ) = 7 x ^ { 3 } + x y - y ^ { 3 }$ at the point $( 4 , - 6 )$ .

Test for relative extrema and saddle points. $z = x ^ { 2 } - 8 x y + y ^ { 2 } + 120 x$

Examine the function $f ( x , y ) = 5 x ^ { 2 } - 8 y ^ { 2 } - 3 x + 8 y + 4$ for relative extrema.

Evaluate the function at the given values of the independent variables. $z = x ^ { 3 } + 2 x y + 3 y ^ { 2 } , \quad x = - 9 , y = 7$

Evaluate the double integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 x + 4 y ) d y d x$ .

Describe the domain and range of the function. $f ( x , y ) = \sqrt { 49 - x ^ { 2 } - y ^ { 2 } }$

Sketch the region $R$ whose area is given by the following double integral. $\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } d y d x$

Describe the trace of the surface given by the function below $y ^ { 2 } + z ^ { 2 } - x ^ { 2 } = 1$ in the xz-plane.

Evaluate the double integral $\int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } x y e ^ { - \left( 4 x ^ { 2 } + 9 y ^ { 2 } \right) } d x d y$ .

Set up an integral for both orders of integration, and use the more convenient order to evaluate the integral below over the region R. $\iint _ { R } \frac { y } { x ^ { 2 } + y ^ { 2 } } d A$ $R$ : trianglebounded by $y = 5 x , y = 6 x$ , and $x = 5$

Evaluate the function $w = \frac { x ^ { 2 } - y z } { x y z }$ at $( - 8,6,3 )$

Use the function $q \left( p _ { 1 } , p _ { 2 } \right) = \frac { 7 p _ { 1 } - 9 p _ { 2 } } { - 9 p _ { 1 } + 9 p _ { 2 } }$ to find $q ( - 5 , - 8 )$