Question 1

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Accepted Answer

A) Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 1 
B) Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 3 
C) Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 0 
D) Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 1 
A) Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 1 
B) Vertical asymptotes: x = -3,x = -2; horizontal asymptote: y = 3 
C) Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 0 
D) Vertical asymptotes: x = -3,x = -2,x = 3; horizontal asymptote: y = 1 A

Question 2

Find constants a,b,and c so that the graph of the function $f ( x ) = c x ^ { 2 } + b x + c$ has a relative maximum at (7,34)and crosses the y-axis at (0,6).

Accepted Answer

A)  $a = \frac { 4 } { 7 }$ ,b = 8,c = 6 
B)  $a = - \frac { 4 } { 7 }$ ,b = -8,c = -6 
C)  $a = \frac { 4 } { 7 }$ ,b = -8,c = 6 
D)  $a = - \frac { 4 } { 7 }$ ,b = 8,c = 6 
A)  $a = \frac { 4 } { 7 }$ ,b = 8,c = 6 
B)  $a = - \frac { 4 } { 7 }$ ,b = -8,c = -6 
C)  $a = \frac { 4 } { 7 }$ ,b = -8,c = 6 
D)  $a = - \frac { 4 } { 7 }$ ,b = 8,c = 6 D

Question 3

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

Accepted Answer

A)  $( 0,0 ) , \left( 1 , \frac { 1 } { 2 } \right) , \left( - 1 , - \frac { 1 } { 2 } \right)$ 
B) (0,0) 
C)  $( 0,0 ) , \left( - 1 , - \frac { 1 } { 2 } \right)$ 
D)  $\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$ 
A)  $( 0,0 ) , \left( 1 , \frac { 1 } { 2 } \right) , \left( - 1 , - \frac { 1 } { 2 } \right)$ 
B) (0,0) 
C)  $( 0,0 ) , \left( - 1 , - \frac { 1 } { 2 } \right)$ 
D)  $\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$ D

Question 4

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

Accepted Answer

A) For x > 2 B) For x -2 D) For x < 2 A) For x > 2 B) For x -2 D) For x < 2

Question 5

An open box is to be made from a square piece of cardboard,36 inches by 36 inches,by removing a small square from each corner and folding up the flaps to form the sides.What are the dimensions of the box of greatest volume that can be constructed in this way?

Accepted Answer

A) 24 inches long by 24 inches wide by 6 inches deep 
B) 6 inches long by 6 inches wide by 24 inches deep 
C) 12 inches long by 12 inches wide by 6 inches deep 
D) 18 inches long by 18 inches wide by 6 inches deep 
A) 24 inches long by 24 inches wide by 6 inches deep 
B) 6 inches long by 6 inches wide by 24 inches deep 
C) 12 inches long by 12 inches wide by 6 inches deep 
D) 18 inches long by 18 inches wide by 6 inches deep

Question 6

Determine the critical numbers of the given function and classify each critical point as a relative maximum,a relative minimum,or neither.Round numbers to two decimal places,if necessary. $f ( x ) = \frac { 1 } { x ^ { 2 } - 8 x + 7 }$

Accepted Answer

A) x = 1,4,7; (1,1)relative maximum; (4,-0.11)relative minimum;(7,1)relative maximum 
B) x = 1,7; (1,1)relative maximum; (7,1)relative maximum 
C) x = 1,7; (1,1)relative maximum; (7,1)relative minimum 
D) x = 1,4,7; (4,-0.11)relative maximum 
A) x = 1,4,7; (1,1)relative maximum; (4,-0.11)relative minimum;(7,1)relative maximum 
B) x = 1,7; (1,1)relative maximum; (7,1)relative maximum 
C) x = 1,7; (1,1)relative maximum; (7,1)relative minimum 
D) x = 1,4,7; (4,-0.11)relative maximum

Question 7

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up.

Accepted Answer

A) Everywhere 
B) For x  3 
C) Nowhere 
D) For -3 < x < 3 
A) Everywhere 
B) For x  3 
C) Nowhere 
D) For -3 < x < 3

Question 8

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } - 18 x + 1$ .

Accepted Answer

A) Relative maximum at (-3,-8); relative minimum at (1,55) 
B) Relative maximum at (-3,55); relative minimum at (1,-9) 
C) Relative maximum at (-3,-9); relative minimum at (-1,23) 
D) Relative maximum at (1,55); relative minimum at (-3,-9) 
A) Relative maximum at (-3,-8); relative minimum at (1,55) 
B) Relative maximum at (-3,55); relative minimum at (1,-9) 
C) Relative maximum at (-3,-9); relative minimum at (-1,23) 
D) Relative maximum at (1,55); relative minimum at (-3,-9)

Question 9

Determine the critical points of the given function and classify each critical point as a relative maximum,a relative minimum,or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

Accepted Answer

A) (0,2)neither; (1,3)relative minimum 
B) (0,2)relative minimum; (1,3)relative maximum 
C) (0,3)relative minimum; (1,4)neither 
D) (0,2)relative minimum; (1,4)neither 
A) (0,2)neither; (1,3)relative minimum 
B) (0,2)relative minimum; (1,3)relative maximum 
C) (0,3)relative minimum; (1,4)neither 
D) (0,2)relative minimum; (1,4)neither

Question 10

Find two non-negative numbers x and y whose sum is 14 and are such that the xy2 is as large as possible.

Accepted Answer

A) 7 and 7 
B)  $\frac { 14 } { 3 }$ and $\frac { 28 } { 3 }$ 
C) 1 and 13 
D) 14 and 28 
A) 7 and 7 
B)  $\frac { 14 } { 3 }$ and $\frac { 28 } { 3 }$ 
C) 1 and 13 
D) 14 and 28

Question 11

The owner of a novelty store can obtain joy buzzers from the manufacturer for 50 cents apiece.He estimates he can sell 60 buzzers when he charges $1.00 apiece for them and that he will be able to sell 10 more buzzers for every 8 cent decrease in price.What price should he charge in order to maximize profit?

Accepted Answer

A) $0.79 
B) $0.89 
C) $1.09 
D) $0.99 
A) $0.79 
B) $0.89 
C) $1.09 
D) $0.99

Question 12

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 5 x + 5$ dollars.If the price is p(x)= (41 - x)dollars per unit,determine the level of production that maximizes profit.

Accepted Answer

A) x = 3 
B) x = 1 
C) x = 5 
D) x = 2 
A) x = 3 
B) x = 1 
C) x = 5 
D) x = 2

Question 13

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ .

Accepted Answer

A) Increasing on x  2,decreasing on -3  2 
C) Increasing on x  3 
D) Increasing on x  $\leq$ -2 and x  $\geq$ 3 ,decreasing on - 2  $\leq$ x  $\leq$ 3 
A) Increasing on x  2,decreasing on -3  2 
C) Increasing on x  3 
D) Increasing on x  $\leq$ -2 and x  $\geq$ 3 ,decreasing on - 2  $\leq$ x  $\leq$ 3

Question 14

A 5-year projection of population trends suggests that t years from now,the population of a certain community will be $P ( t ) = - t ^ { 3 } + 15 t ^ { 2 } + 72 t + 55$ thousand. 
a.At what time during the 5-year period will the population be growing most rapidly?
b. At what time during the 5-year period will the population be growing least rapidly?
c. At what time is the rate of population growth changing most rapidly?

Accepted Answer

A) a.t = 2 yearsb.t = 0 yearsc.t = 2 years 
B) a.t = 2 yearsb.t = 1 yearc.t = 3 years 
C) a.t = 5 yearsb.t = 0 yearsc.t = 0 years 
D) a.t = 0 yearsb.t = 0 yearsc.t = 2 years 
A) a.t = 2 yearsb.t = 0 yearsc.t = 2 years 
B) a.t = 2 yearsb.t = 1 yearc.t = 3 years 
C) a.t = 5 yearsb.t = 0 yearsc.t = 0 years 
D) a.t = 0 yearsb.t = 0 yearsc.t = 2 years

Question 15

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1  $\leq$ x  $\leq$ 1.

Accepted Answer

A) -2 
B) 0 
C) 1 
D) -1 
A) -2 
B) 0 
C) 1 
D) -1

Question 16

Name the vertical and horizontal asymptotes of the given graph.

Accepted Answer

A) Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptotes: y = 5,y = 0 
B) Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptote: y = 5 
C) Vertical asymptotes: x = -2,x = 3; horizontal asymptote: y = 5 
D) Vertical asymptotes: x = -2,x = 3; horizontal asymptotes: y = 5,y = 0 
A) Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptotes: y = 5,y = 0 
B) Vertical asymptotes: x = -2,x = 3,x = 0; horizontal asymptote: y = 5 
C) Vertical asymptotes: x = -2,x = 3; horizontal asymptote: y = 5 
D) Vertical asymptotes: x = -2,x = 3; horizontal asymptotes: y = 5,y = 0

Question 17

Sketch the graph of the given function. $f ( x ) = \frac { x + 5 } { x + 2 }$

Accepted Answer

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

Question 18

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 80 x - x ^ { 2 } } { x ^ { 2 } + 80 }$ million dollars.What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places,if necessary.

Accepted Answer

A) Maximum at x = 8 and maximum revenue is R(8)= 4 (million dollars) 
B) Maximum at x = 8 and maximum revenue is R(8)= 32 (million dollars) 
C) Maximum at x = 9 and maximum revenue is R(9)= 2.67 (million dollars) 
D) Maximum at x = 8 and maximum revenue is R(8)= 2.67 (thousand dollars) 
A) Maximum at x = 8 and maximum revenue is R(8)= 4 (million dollars) 
B) Maximum at x = 8 and maximum revenue is R(8)= 32 (million dollars) 
C) Maximum at x = 9 and maximum revenue is R(9)= 2.67 (million dollars) 
D) Maximum at x = 8 and maximum revenue is R(8)= 2.67 (thousand dollars)

Question 19

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1  $\leq$ x  $\leq$ 5.

Accepted Answer

A) 5 
B) -4 
C) -1 
D) 0 
A) 5 
B) -4 
C) -1 
D) 0

Question 20

The demand function for a certain commodity is $D ( p ) = \frac { 30 } { p + 5 }$ .For what values of p is the demand inelastic?

Accepted Answer

A) p > -5 B) p > 0 C) p < -5 D) p < 0 A) p > -5 B) p > 0 C) p < -5 D) p < 0

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Find constants a,b,and c so that the graph of the function $f ( x ) = c x ^ { 2 } + b x + c$ has a relative maximum at (7,34)and crosses the y-axis at (0,6).

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

An open box is to be made from a square piece of cardboard,36 inches by 36 inches,by removing a small square from each corner and folding up the flaps to form the sides.What are the dimensions of the box of greatest volume that can be constructed in this way?

Determine the critical numbers of the given function and classify each critical point as a relative maximum,a relative minimum,or neither.Round numbers to two decimal places,if necessary. $f ( x ) = \frac { 1 } { x ^ { 2 } - 8 x + 7 }$

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up.

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } - 18 x + 1$ .

Determine the critical points of the given function and classify each critical point as a relative maximum,a relative minimum,or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

Find two non-negative numbers x and y whose sum is 14 and are such that the xy² is as large as possible.

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 5 x + 5$ dollars.If the price is p(x)= (41 - x)dollars per unit,determine the level of production that maximizes profit.

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\leq$ x $\leq$ 1.

Name the vertical and horizontal asymptotes of the given graph.

Sketch the graph of the given function. $f ( x ) = \frac { x + 5 } { x + 2 }$

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 80 x - x ^ { 2 } } { x ^ { 2 } + 80 }$ million dollars.What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places,if necessary.

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\leq$ x $\leq$ 5.

The demand function for a certain commodity is $D ( p ) = \frac { 30 } { p + 5 }$ .For what values of p is the demand inelastic?

Functions,graphs,and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

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Exam 3: Additional Applications of the Derivative

Find all vertical and horizontal asymptotes of the graph of the given function. f(x)=x2−3xx2+5x+6f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }f(x)=x2+5x+6x2−3x​

Find constants a,b,and c so that the graph of the function f(x)=cx2+bx+cf ( x ) = c x ^ { 2 } + b x + cf(x)=cx2+bx+c has a relative maximum at (7,34)and crosses the y-axis at (0,6).

Locate all inflection points of f(x)=xx2+1f ( x ) = \frac { x } { x ^ { 2 } + 1 }f(x)=x2+1x​ .

Determine where the graph of f(x)=x3−6x2+8x+4f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4f(x)=x3−6x2+8x+4 is concave down.

An open box is to be made from a square piece of cardboard,36 inches by 36 inches,by removing a small square from each corner and folding up the flaps to form the sides.What are the dimensions of the box of greatest volume that can be constructed in this way?

Determine the critical numbers of the given function and classify each critical point as a relative maximum,a relative minimum,or neither.Round numbers to two decimal places,if necessary. f(x)=1x2−8x+7f ( x ) = \frac { 1 } { x ^ { 2 } - 8 x + 7 }f(x)=x2−8x+71​

Determine where the graph of f(x)=x4−54x2+6f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6f(x)=x4−54x2+6 is concave up.

Use the second derivative test to find the relative maxima and minima of the function f(x)=2x3+6x2−18x+1f ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } - 18 x + 1f(x)=2x3+6x2−18x+1 .

Determine the critical points of the given function and classify each critical point as a relative maximum,a relative minimum,or neither. f(x)=3x4−8x3+6x2+3f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3f(x)=3x4−8x3+6x2+3

Find two non-negative numbers x and y whose sum is 14 and are such that the xy2 is as large as possible.

The cost of producing x units of a certain commodity is C(x)=3x2+5x+5C ( x ) = 3 x ^ { 2 } + 5 x + 5C(x)=3x2+5x+5 dollars.If the price is p(x)= (41 - x)dollars per unit,determine the level of production that maximizes profit.

Find the intervals of increase and decrease for f(x)=6x3+9x2−108x−3f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3f(x)=6x3+9x2−108x−3 .

Find the absolute maximum of the function f(x)=x7−x6f ( x ) = x ^ { 7 } - x ^ { 6 }f(x)=x7−x6 on the interval -1 ≤\leq≤ x ≤\leq≤ 1.

Name the vertical and horizontal asymptotes of the given graph.

Sketch the graph of the given function. f(x)=x+5x+2f ( x ) = \frac { x + 5 } { x + 2 }f(x)=x+2x+5​

Find the absolute minimum of the function f(x)=x3−3x2f ( x ) = x ^ { 3 } - 3 x ^ { 2 }f(x)=x3−3x2 on the interval 1 ≤\leq≤ x ≤\leq≤ 5.

The demand function for a certain commodity is D(p)=30p+5D ( p ) = \frac { 30 } { p + 5 }D(p)=p+530​ .For what values of p is the demand inelastic?

Functions,graphs,and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Itegration

Additional Topics in Integration

Calculus of Several Variables

Differential Equations

Infinite Series and Taylor Series Approximations

Probability and Calculus

Trigonometric Functions

Calculus Practice Questions

Mathematical Problem Set: Calculus, Algebra, and Optimization

Filters

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Find constants a,b,and c so that the graph of the function $f ( x ) = c x ^ { 2 } + b x + c$ has a relative maximum at (7,34)and crosses the y-axis at (0,6).

Locate all inflection points of $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ .

Determine where the graph of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 8 x + 4$ is concave down.

Determine the critical numbers of the given function and classify each critical point as a relative maximum,a relative minimum,or neither.Round numbers to two decimal places,if necessary. $f ( x ) = \frac { 1 } { x ^ { 2 } - 8 x + 7 }$

Determine where the graph of $f ( x ) = x ^ { 4 } - 54 x ^ { 2 } + 6$ is concave up.

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 2 x ^ { 3 } + 6 x ^ { 2 } - 18 x + 1$ .

Determine the critical points of the given function and classify each critical point as a relative maximum,a relative minimum,or neither. $f ( x ) = 3 x ^ { 4 } - 8 x ^ { 3 } + 6 x ^ { 2 } + 3$

Find two non-negative numbers x and y whose sum is 14 and are such that the xy² is as large as possible.

The cost of producing x units of a certain commodity is $C ( x ) = 3 x ^ { 2 } + 5 x + 5$ dollars.If the price is p(x)= (41 - x)dollars per unit,determine the level of production that maximizes profit.

Find the intervals of increase and decrease for $f ( x ) = 6 x ^ { 3 } + 9 x ^ { 2 } - 108 x - 3$ .

Find the absolute maximum of the function $f ( x ) = x ^ { 7 } - x ^ { 6 }$ on the interval -1 $\leq$ x $\leq$ 1.

Sketch the graph of the given function. $f ( x ) = \frac { x + 5 } { x + 2 }$

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval 1 $\leq$ x $\leq$ 5.

The demand function for a certain commodity is $D ( p ) = \frac { 30 } { p + 5 }$ .For what values of p is the demand inelastic?