Question 1

Evaluate the following double integral: $\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$

Accepted Answer

A) 4 
B) The integral can't be evaluated. 
C)  $\frac { 1 } { 4 }$ 
D) 0 
A) 4 
B) The integral can't be evaluated. 
C)  $\frac { 1 } { 4 }$ 
D) 0 D

Question 2

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 18 cents per can and a well-known national brand that it obtains at the cost of 40 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

A) local brand (x)at 39 cents,national brand (y)at 104 cents 
B) local brand (x)at 46 cents,national brand (y)at 52 cents 
C) local brand (x)at 44 cents,national brand (y)at 52 cents 
D) local brand (x)at 88 cents,national brand (y)at 50 cents 
A) local brand (x)at 39 cents,national brand (y)at 104 cents 
B) local brand (x)at 46 cents,national brand (y)at 52 cents 
C) local brand (x)at 44 cents,national brand (y)at 52 cents 
D) local brand (x)at 88 cents,national brand (y)at 50 cents C

Question 3

Use a double integral to find the area of R. R is the triangle with vertices (-2,1),(2,1),and (0,-1).

Accepted Answer

A)  $\int _ { x - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 4$ 
B)  $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 4$ 
C)  $\int _ { x = - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 8$ 
D)  $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 8$ 
A)  $\int _ { x - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 4$ 
B)  $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 4$ 
C)  $\int _ { x = - 2 } ^ { x - 2 } \int _ { y - x - 1 } ^ { y - x - 1 } 1 d y d x = 8$ 
D)  $\int _ { y - 1 } ^ { y - 1 } \int _ { x - - y - 1 } ^ { x - y + 1 } 1 d x d y = 8$ B

Question 4

Compute f (ln 2,ln 5)if $f ( x , y ) = e ^ { 2 x + y }$ .

Accepted Answer

A) 10 
B) 9 
C) 20 
D) None of the above 
A) 10 
B) 9 
C) 20 
D) None of the above

Question 5

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

Accepted Answer

A)  $7 y ^ { 6 }$ 
B)  $42 x y ^ { 5 } + 7 y ^ { 6 }$ 
C) 7x 
D)  $42 x y ^ { 5 }$ 
A)  $7 y ^ { 6 }$ 
B)  $42 x y ^ { 5 } + 7 y ^ { 6 }$ 
C) 7x 
D)  $42 x y ^ { 5 }$

Question 6

Use Lagrange multipliers to find the maximum value of f (x,y)= 9xy subject to the constraint 9x + 3y = 27.

Accepted Answer

A) f (0,0)= 0 
B) f (3,9)= 243 
C)  $f \left( \frac { 3 } { 4 } , \frac { 9 } { 4 } \right) = \frac { 243 } { 16 }$ 
D)  $f \left( \frac { 3 } { 2 } , \frac { 9 } { 2 } \right) = \frac { 243 } { 4 }$ 
A) f (0,0)= 0 
B) f (3,9)= 243 
C)  $f \left( \frac { 3 } { 4 } , \frac { 9 } { 4 } \right) = \frac { 243 } { 16 }$ 
D)  $f \left( \frac { 3 } { 2 } , \frac { 9 } { 2 } \right) = \frac { 243 } { 4 }$

Question 7

Given the following points in the plane,find the slope of the least squares line: (1,2),(2,1),(3,3),and (5,6) round your answer to two decimal places,if necessary.

Accepted Answer

A) 0.97 
B) 1.03 
C) 1.14 
D) 1.09 
A) 0.97 
B) 1.03 
C) 1.14 
D) 1.09

Question 8

The following data shows the age and income for a small number of people. 
$$\begin{array} { r l l l l l } 
\text { Age (years) } & 24 & 32 & 39 & 54 & 60 \
\text { Incomes (\$) } & 30,000 & 34,000 & 53,000 & 72,000 & 81,000
\end{array}$$
Find the best fit straight line of this data,rounding coefficients and constants to the nearest whole number.Let x represent age and y represent income.

Accepted Answer

A) y = 3,830x - 8,036 
B) y = 3,106x - 8,048 
C) y = 1,484x - 8,041 
D) None of the above 
A) y = 3,830x - 8,036 
B) y = 3,106x - 8,048 
C) y = 1,484x - 8,041 
D) None of the above

Question 9

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

Accepted Answer

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

Question 10

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 345 miles away?

Accepted Answer

A) 10.4 minutes 
B) 13.7 minutes 
C) 12.9 minutes 
D) 11.5 minutes 
A) 10.4 minutes 
B) 13.7 minutes 
C) 12.9 minutes 
D) 11.5 minutes

Question 11

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

Question 12

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

Accepted Answer

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

Question 13

Compute f (6,0)if $f ( x , y ) = \frac { x } { e ^ { y } }$ .

Accepted Answer

A) Undefined 
B) 6 
C) 0 
D)  $\frac { 1 } { 6 }$ 
A) Undefined 
B) 6 
C) 0 
D)  $\frac { 1 } { 6 }$

Question 14

A manufacturer is planning to sell a new product at the price of $230 per unit and estimates that if x thousand dollars is spent on development and y thousand dollars is spent on promotion,approximately $\frac { 340 y } { y + 7 } + \frac { 170 x } { x + 14 }$ units of the product will be sold.The cost of manufacturing the product is $160 per unit.If the manufacturer has a total of $340,000 to spend on development and promotion,how should this money be allocated to generate the largest possible profit? [Hint: Profit equals (number of units)(price per unit minus cost per unit)minus total amount spent on development and promotion.]

Accepted Answer

A) $173,500 on development,$167,000 on promotion 
B) $167,000 on development,$173,000 on promotion 
C) $166,500 on development,$173,500 on promotion 
D) $173,000 on development,$167,000 on promotion 
A) $173,500 on development,$167,000 on promotion 
B) $167,000 on development,$173,000 on promotion 
C) $166,500 on development,$173,500 on promotion 
D) $173,000 on development,$167,000 on promotion

Question 15

Given the function of three variables f (x,y,z)= xy + xz + yz,evaluate f (3,7,7).

Accepted Answer

A) 86 
B) 100 
C) 91 
D) 94 
A) 86 
B) 100 
C) 91 
D) 94

Question 16

Find the maximum value of the function f (x,y,z)= 7x + 8y + 6z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 596$ .

Accepted Answer

A) 298 
B) 894 
C) 1,043.0 
D) 149 
A) 298 
B) 894 
C) 1,043.0 
D) 149

Question 17

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$

Accepted Answer

A)  $f _ { x } = 16 ( 4 x + 2 y ) ^ { 5 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 5 }$ 
B)  $f _ { x } = 16 ( x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 ) ^ { 3 }$ 
C)  $f _ { x } = 16 ( 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x ) ^ { 3 }$ 
D)  $f _ { x } = 16 ( 4 x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 3 }$ 
A)  $f _ { x } = 16 ( 4 x + 2 y ) ^ { 5 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 5 }$ 
B)  $f _ { x } = 16 ( x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 ) ^ { 3 }$ 
C)  $f _ { x } = 16 ( 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x ) ^ { 3 }$ 
D)  $f _ { x } = 16 ( 4 x + 2 y ) ^ { 3 }$ , $f _ { y } = 8 ( 4 x + 2 y ) ^ { 3 }$

Question 18

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = ex,y = 2,and x = 0

Accepted Answer

A) Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 2 \
0 & \leq y \leq \ln 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 2 \
1 \leq x \leq 2
\end{array}$ 
B) Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq \ln 2 \
0 & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 2 \
1 \leq x \leq \ln 2
\end{array}$ 
C) Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
1 \leq y \leq 2 \
0 \leq x \leq \ln 2
\end{array}$ 
D) Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 2 \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
1 \leq y \leq 2 \
0 \leq x \leq \ln y
\end{array}$ 
A) Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq 2 \
0 & \leq y \leq \ln 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq \ln 2 \
1 \leq x \leq 2
\end{array}$ 
B) Vertical cross sections: $\begin{aligned}
e ^ { x } & \leq x \leq \ln 2 \
0 & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
0 \leq y \leq 2 \
1 \leq x \leq \ln 2
\end{array}$ 
C) Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln y \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
1 \leq y \leq 2 \
0 \leq x \leq \ln 2
\end{array}$ 
D) Vertical cross sections: $\begin{aligned}
0 & \leq x \leq \ln 2 \
e ^ { x } & \leq y \leq 2
\end{aligned}$ Horizontal cross sections: $\begin{array} { l } 
1 \leq y \leq 2 \
0 \leq x \leq \ln y
\end{array}$

Question 19

At a certain factory,the output Q is related to inputs x and y by the expression $Q ( x , y ) = 2 x ^ { 3 } + 2 x ^ { 2 } y ^ { 4 } + 9 y ^ { 5 }$ .If 0  $\leq$ x  $\leq$ 8 and 0  $\leq$ y  $\leq$ 10,what is the average output of the factory? (Approximate to 2 decimal places.)

Accepted Answer

A) 111,504.00 units 
B) 235,589.33 units 
C) 235,590.13 units 
D) 308,341.33 units 
A) 111,504.00 units 
B) 235,589.33 units 
C) 235,590.13 units 
D) 308,341.33 units

Question 20

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 (9,8)

Accepted Answer

A) -0.90 
B) -1.50 
C) 0.60 
D) 0.72 
A) -0.90 
B) -1.50 
C) 0.60 
D) 0.72

Evaluate the following double integral: $\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$

Use a double integral to find the area of R. R is the triangle with vertices (-2,1),(2,1),and (0,-1).

Compute f (ln 2,ln 5)if $f ( x , y ) = e ^ { 2 x + y }$ .

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

Use Lagrange multipliers to find the maximum value of f (x,y)= 9xy subject to the constraint 9x + 3y = 27.

Given the following points in the plane,find the slope of the least squares line: (1,2),(2,1),(3,3),and (5,6) round your answer to two decimal places,if necessary.

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

A military radar is measuring the distance to a jet fighter.The radar has received the following measurements: time t (minutes) 1 2 3 4 5 6 distance (miles) 280 289 294 299 308 310 Using a least squares fit to the data,extrapolate to the nearest tenth of a minute when the jet will be 345 miles away?

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 $f ( x , y ) = e ^ { 8 x y }$

Compute f (6,0)if $f ( x , y ) = \frac { x } { e ^ { y } }$ .

Given the function of three variables f (x,y,z)= xy + xz + yz,evaluate f (3,7,7).

Find the maximum value of the function f (x,y,z)= 7x + 8y + 6z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 596$ .

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = e^x,y = 2,and x = 0

At a certain factory,the output Q is related to inputs x and y by the expression $Q ( x , y ) = 2 x ^ { 3 } + 2 x ^ { 2 } y ^ { 4 } + 9 y ^ { 5 }$ .If 0 $\leq$ x $\leq$ 8 and 0 $\leq$ y $\leq$ 10,what is the average output of the factory? (Approximate to 2 decimal places.)

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 (9,8)

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Exam 7: Calculus of Several Variables

Evaluate the following double integral: ∫−47∫−44x4y7dydx\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x∫−47​∫−44​x4y7dydx

Use a double integral to find the area of R. R is the triangle with vertices (-2,1),(2,1),and (0,-1).

Compute f (ln 2,ln 5)if f(x,y)=e2x+yf ( x , y ) = e ^ { 2 x + y }f(x,y)=e2x+y .

Compute fyf _ { y }fy​ for f(x,y)=7xy6f ( x , y ) = 7 x y ^ { 6 }f(x,y)=7xy6

Use Lagrange multipliers to find the maximum value of f (x,y)= 9xy subject to the constraint 9x + 3y = 27.

Given the following points in the plane,find the slope of the least squares line: (1,2),(2,1),(3,3),and (5,6) round your answer to two decimal places,if necessary.

Compute fy for f(x,y)=e8xyf _ { y } \text { for } f ( x , y ) = e ^ { 8 x y }fy​ for f(x,y)=e8xy

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)=e8xyf ( x , y ) = e ^ { 8 x y }f(x,y)=e8xy

Compute f (6,0)if f(x,y)=xeyf ( x , y ) = \frac { x } { e ^ { y } }f(x,y)=eyx​ .

Given the function of three variables f (x,y,z)= xy + xz + yz,evaluate f (3,7,7).

Find the maximum value of the function f (x,y,z)= 7x + 8y + 6z on the sphere x2+y2+z2=596x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 596x2+y2+z2=596 .

Compute all first-order partial derivatives of the given function. f(x,y)=(4x+2y)4f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }f(x,y)=(4x+2y)4

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = ex,y = 2,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 (9,8)

Functions,graphs,and Limits

Differentiation: Basic Concepts

Additional Applications of the Derivative

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Evaluate the following double integral: $\int _ { - 4 } ^ { 7 } \int _ { - 4 } ^ { 4 } x ^ { 4 } y ^ { 7 } d y d x$

Compute f (ln 2,ln 5)if $f ( x , y ) = e ^ { 2 x + y }$ .

Compute $f _ { y }$ for $f ( x , y ) = 7 x y ^ { 6 }$

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

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 $f ( x , y ) = e ^ { 8 x y }$

Compute f (6,0)if $f ( x , y ) = \frac { x } { e ^ { y } }$ .

Find the maximum value of the function f (x,y,z)= 7x + 8y + 6z on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 596$ .

Compute all first-order partial derivatives of the given function. $f ( x , y ) = ( 4 x + 2 y ) ^ { 4 }$

Use inequalities to describe R in terms of its vertical and horizontal cross sections. R is the region bounded by y = e^x,y = 2,and x = 0