Question 1

Compute $f _ { x }$ for $f ( x , y ) = e ^ { xy }$ .

Accepted Answer

A)  $e ^ { x y }$ 
B)  $x e ^ { x y }$ 
C)  $y e ^ { xy }$ 
D)  $x y e ^ { x y }$ 
A)  $e ^ { x y }$ 
B)  $x e ^ { x y }$ 
C)  $y e ^ { xy }$ 
D)  $x y e ^ { x y }$

Question 2

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 14 x + y ^ { 2 } - 6 y$ .

Accepted Answer

Relative m

Question 3

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = ex, y = 4, and x = 0.

Accepted Answer

A)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 4 \
e ^ { x } & \leq y \leq 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 4 \
0 \leq x \leq \ln y
\end{array}$ 
B)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 4 \
0 \leq x \leq \ln 4
\end{array}$ 
C)  Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 4 \
0 & \leq y \leq \ln 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 4 \
1 \leq x \leq 4
\end{array}$ 
D)  Vertical cross sections: $\begin{array} { c } 
e ^ { x } \leq x \leq \ln 4 \
0 \leq y \leq 4
\end{array}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 4 \
1 \leq x \leq \ln 4
\end{array}$ 
A)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 4 \
e ^ { x } & \leq y \leq 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 4 \
0 \leq x \leq \ln y
\end{array}$ 
B)  Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { c } 
1 \leq y \leq 4 \
0 \leq x \leq \ln 4
\end{array}$ 
C)  Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 4 \
0 & \leq y \leq \ln 4
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 4 \
1 \leq x \leq 4
\end{array}$ 
D)  Vertical cross sections: $\begin{array} { c } 
e ^ { x } \leq x \leq \ln 4 \
0 \leq y \leq 4
\end{array}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 4 \
1 \leq x \leq \ln 4
\end{array}$

Question 4

Given the following points in the plane, find the y-intercept of the least squares line to two decimal places: (3, 3), (5, 5), (7, 8), and (9, 13)

Accepted Answer

The answer of Given the following points in the plane,...

Question 5

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Accepted Answer

The answer of Find the maximum value of f (x,...

Question 6

Find the volume of the solid bounded above by the graph of the function f (x, y) = xy and below by the rectangular region R defined by: 0 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 4.

Accepted Answer

The answer of Find the volume of the solid bounded...

Question 7

The only grocery store in a small rural community carries two brands of frozen apple juice, a local brand that it obtains at the cost of 22 cents per can and a well-known national brand that it obtains at the cost of 60 cents per can. The grocer estimates that if the local brand is sold for x cents per can and the national brand for y cents per can, approximately 70 - 5x + 4y cans of the local brand and 80 + 6x - 7y cans of the national brand will be sold each day. How should the grocer price each brand to maximize the profit from the sale of the juice?

Accepted Answer

local brand (x) at 4

Question 8

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1).

Accepted Answer

A)  
$\frac { 68 } { 3 }$ 
B)  
$\frac { 94 } { 3 }$ 
C)  12 
D)  18 
A)  
$\frac { 68 } { 3 }$ 
B)  
$\frac { 94 } { 3 }$ 
C)  12 
D)  18

Question 9

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 4 x y }$

Accepted Answer

The answer of Compute $f _ { x }$ for...

Question 10

The accompanying table lists the high-school GPA and college GPA for a number of students $$\begin{array} { l l l l l l l } 
\text { High school GPA } & 2.0 & 2.5 & 3.1 & 3.7 & 3.7 & 4.0 \
\text { College GPA } & 3.2 & 2.6 & 3.0 & 3.6 & 3.8 & 3.7
\end{array}$$
 Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

Accepted Answer

A)  3.4 
B)  3.3 
C)  3.6 
D)  3.5 
A)  3.4 
B)  3.3 
C)  3.6 
D)  3.5

Question 11

Find the minimum value of f (x, y) = x + 2y subject to the constraint $g ( x , y ) = x y ^ { 2 } = 8$ .

Accepted Answer

The answer of Find the minimum value of f (x,...

Question 12

Compute $f _ { y }$ for $f ( x , y ) = 2 x y ^ { 8 }$ .

Accepted Answer

A)  $16 x y ^ { 7 }$ 
B)  2x 
C)  $2 y ^ { 8 }$ 
D)  $16 x y ^ { 7 } + 2 y ^ { 8 }$ 
A)  $16 x y ^ { 7 }$ 
B)  2x 
C)  $2 y ^ { 8 }$ 
D)  $16 x y ^ { 7 } + 2 y ^ { 8 }$

Question 13

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 2 x y }$ .

Accepted Answer

A)  $2 y e ^ { 2 x y }$ 
B)  $2 x e ^ { 2 xy }$ 
C)  $2 e ^ { 2 x y }$ 
D)  $2 x y e ^ { 2 x y }$ 
A)  $2 y e ^ { 2 x y }$ 
B)  $2 x e ^ { 2 xy }$ 
C)  $2 e ^ { 2 x y }$ 
D)  $2 x y e ^ { 2 x y }$

Question 14

A military radar is measuring the distance to a jet fighter. The radar has received the following measurements: $$\begin{array} { l l l l l l l } 
\text { time } t \text { (minutes) } & 1 & 2 & 3 & 4 & 5 & 6 \
\text { distance (miles) } & 280 & 289 & 294 & 299 & 308 & 310
\end{array}$$ 
Using a least squares fit to the data, extrapolate to the nearest tenth of a minute when the jet will be 394 miles away?

Accepted Answer

The answer of A military radar is measuring the distance...

Question 15

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 5 x - 5 y$ .

Accepted Answer

Saddle poi

Question 16

Find the maximum value of the function f (x, y, z) = 4x + 5y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 360$ .

Accepted Answer

A)  630 
B)  480.0 
C)  90 
D)  180 
A)  630 
B)  480.0 
C)  90 
D)  180

Question 17

Let f (x, y) = 7xy. Compute f (5, 0).

Accepted Answer

The answer of Let f (x, y) = 7xy. Compute...

Question 18

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 8 xy }$

Accepted Answer

A)  $64 e ^ { 8 x y }$ 
B)  $64 x ^ { 2 } e ^ { 8 x y }$ 
C)  $64 x ^ { 2 } y ^ { 2 } e ^ { 8 x y }$ 
D)  $64 y ^ { 2 } e ^ { 8 xy }$ 
A)  $64 e ^ { 8 x y }$ 
B)  $64 x ^ { 2 } e ^ { 8 x y }$ 
C)  $64 x ^ { 2 } y ^ { 2 } e ^ { 8 x y }$ 
D)  $64 y ^ { 2 } e ^ { 8 xy }$

Question 19

Find the maximum value of the function f (x, y, z) = 5x + 6y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 440$

Accepted Answer

The answer of Find the maximum value of the function...

Question 20

If $f ( x , y ) = x e ^ { xy }$ , find $f _ { y } ( 6,2 )$ .

Accepted Answer

The answer of If \(f ( x , y )...

Compute $f _ { x }$ for $f ( x , y ) = e ^ { xy }$ .

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 14 x + y ^ { 2 } - 6 y$ .

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = e^x, y = 4, and x = 0.

Given the following points in the plane, find the y-intercept of the least squares line to two decimal places: (3, 3), (5, 5), (7, 8), and (9, 13)

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Find the volume of the solid bounded above by the graph of the function f (x, y) = xy and below by the rectangular region R defined by: 0 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 4.

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1).

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 4 x y }$

The accompanying table lists the high-school GPA and college GPA for a number of students High school GPA 2.0 2.5 3.1 3.7 3.7 4.0 College GPA 3.2 2.6 3.0 3.6 3.8 3.7 Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

Find the minimum value of f (x, y) = x + 2y subject to the constraint $g ( x , y ) = x y ^ { 2 } = 8$ .

Compute $f _ { y }$ for $f ( x , y ) = 2 x y ^ { 8 }$ .

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 2 x y }$ .

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 5 x - 5 y$ .

Find the maximum value of the function f (x, y, z) = 4x + 5y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 360$ .

Let f (x, y) = 7xy. Compute f (5, 0).

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 8 xy }$

Find the maximum value of the function f (x, y, z) = 5x + 6y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 440$

If $f ( x , y ) = x e ^ { xy }$ , find $f _ { y } ( 6,2 )$ .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Appendix: Algebra Review

Filters

Exam 7: Calculus of Several Variables

Compute fxf _ { x }fx​ for f(x,y)=exyf ( x , y ) = e ^ { xy }f(x,y)=exy .

Find the extrema (minima, maxima, saddle points), if any, for f(x,y)=x2−14x+y2−6yf ( x , y ) = x ^ { 2 } - 14 x + y ^ { 2 } - 6 yf(x,y)=x2−14x+y2−6y .

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = ex, y = 4, and x = 0.

Given the following points in the plane, find the y-intercept of the least squares line to two decimal places: (3, 3), (5, 5), (7, 8), and (9, 13)

Find the maximum value of f (x, y) = xy on the ellipse 4x2+9y2=364 x ^ { 2 } + 9 y ^ { 2 } = 364x2+9y2=36 .

Find the volume of the solid bounded above by the graph of the function f (x, y) = xy and below by the rectangular region R defined by: 0 ≤\le≤ x ≤\le≤ 3 and 0 ≤\le≤ y ≤\le≤ 4.

Evaluate the given double integral for the specified region R. ∬R(8x+2y)dA\iint _ { R } ( 8 x + 2 y ) d A∬R​(8x+2y)dA , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1).

Compute fxf _ { x }fx​ for f(x,y)=e4xyf ( x , y ) = e ^ { 4 x y }f(x,y)=e4xy

The accompanying table lists the high-school GPA and college GPA for a number of students High school GPA 2.0 2.5 3.1 3.7 3.7 4.0 College GPA 3.2 2.6 3.0 3.6 3.8 3.7 Using the best fit straight line, predict the college GPA (to one decimal place) for a student whose high school GPA was 3.5.

Find the minimum value of f (x, y) = x + 2y subject to the constraint g(x,y)=xy2=8g ( x , y ) = x y ^ { 2 } = 8g(x,y)=xy2=8 .

Compute fyf _ { y }fy​ for f(x,y)=2xy8f ( x , y ) = 2 x y ^ { 8 }f(x,y)=2xy8 .

Compute fxf _ { x }fx​ for f(x,y)=e2xyf ( x , y ) = e ^ { 2 x y }f(x,y)=e2xy .

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for f(x,y)=x2+xy−y2+5x−5yf ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 5 x - 5 yf(x,y)=x2+xy−y2+5x−5y .

Find the maximum value of the function f (x, y, z) = 4x + 5y + 7z on the sphere x2+y2+z2=360x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 360x2+y2+z2=360 .

Let f (x, y) = 7xy. Compute f (5, 0).

Compute fyyf _ { y y }fyy​ for f(x,y)=e8xyf ( x , y ) = e ^ { 8 xy }f(x,y)=e8xy

Find the maximum value of the function f (x, y, z) = 5x + 6y + 7z on the sphere x2+y2+z2=440x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 440x2+y2+z2=440

If f(x,y)=xexyf ( x , y ) = x e ^ { xy }f(x,y)=xexy , find fy(6,2)f _ { y } ( 6,2 )fy​(6,2) .

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Appendix: Algebra Review

Filters

Compute $f _ { x }$ for $f ( x , y ) = e ^ { xy }$ .

Find the extrema (minima, maxima, saddle points), if any, for $f ( x , y ) = x ^ { 2 } - 14 x + y ^ { 2 } - 6 y$ .

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = e^x, y = 4, and x = 0.

Find the maximum value of f (x, y) = xy on the ellipse $4 x ^ { 2 } + 9 y ^ { 2 } = 36$ .

Find the volume of the solid bounded above by the graph of the function f (x, y) = xy and below by the rectangular region R defined by: 0 $\le$ x $\le$ 3 and 0 $\le$ y $\le$ 4.

Evaluate the given double integral for the specified region R. $\iint _ { R } ( 8 x + 2 y ) d A$ , where R is the triangle with vertices (0, 0), (2, 0), and (0, 1).

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 4 x y }$

Find the minimum value of f (x, y) = x + 2y subject to the constraint $g ( x , y ) = x y ^ { 2 } = 8$ .

Compute $f _ { y }$ for $f ( x , y ) = 2 x y ^ { 8 }$ .

Compute $f _ { x }$ for $f ( x , y ) = e ^ { 2 x y }$ .

Find the critical points and classify each as being a minimum, maximum, saddle point, or other, for $f ( x , y ) = x ^ { 2 } + x y - y ^ { 2 } + 5 x - 5 y$ .

Find the maximum value of the function f (x, y, z) = 4x + 5y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 360$ .

Compute $f _ { y y }$ for $f ( x , y ) = e ^ { 8 xy }$

Find the maximum value of the function f (x, y, z) = 5x + 6y + 7z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 440$

If $f ( x , y ) = x e ^ { xy }$ , find $f _ { y } ( 6,2 )$ .