Question 1

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

Accepted Answer

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

Question 2

A Florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 500 oranges. The average yield will decrease by 5 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield?

Accepted Answer

A)  80 trees 
B)  30 trees 
C)  20 trees 
D)  75 trees 
A)  80 trees 
B)  30 trees 
C)  20 trees 
D)  75 trees

Question 3

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 4

Find all intervals where the derivative of the function shown below is negative.

Accepted Answer

The answer of Find all intervals where the derivative of...

Question 5

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 4 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1$ .

Accepted Answer

A)  Relative maximum at (1, 109); relative minimum at (-3, -19) 
B)  Relative maximum at (-3, -19); relative minimum at (-1, 45) 
C)  Relative maximum at (-3, -18); relative minimum at (1, 109) 
D)  Relative maximum at (-3, 109); relative minimum at (1, -19) 
A)  Relative maximum at (1, 109); relative minimum at (-3, -19) 
B)  Relative maximum at (-3, -19); relative minimum at (-1, 45) 
C)  Relative maximum at (-3, -18); relative minimum at (1, 109) 
D)  Relative maximum at (-3, 109); relative minimum at (1, -19)

Question 6

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 48 x - x ^ { 2 } } { x ^ { 2 } + 48 }$ 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 = 6 and maximum revenue is R(6) = 18 (million dollars) 
B)  Maximum at x = 6 and maximum revenue is R(6) = 2.05 (thousand dollars) 
C)  Maximum at x = 6 and maximum revenue is R(6) = 3 (million dollars) 
D)  Maximum at x = 7 and maximum revenue is R(7) = 2.05 (million dollars) 
A)  Maximum at x = 6 and maximum revenue is R(6) = 18 (million dollars) 
B)  Maximum at x = 6 and maximum revenue is R(6) = 2.05 (thousand dollars) 
C)  Maximum at x = 6 and maximum revenue is R(6) = 3 (million dollars) 
D)  Maximum at x = 7 and maximum revenue is R(7) = 2.05 (million dollars)

Question 7

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).

Accepted Answer

A)  $a = - \frac { 4 } { 5 }$ , b = -8, c = -5 
B)  $a = - \frac { 4 } { 5 }$ , b = 8, c = 5 
C)  $a = \frac { 4 } { 5 }$ , b = 8, c = 5 
D)  $a = \frac { 4 } { 5 }$ , b = -8, c = 5 
A)  $a = - \frac { 4 } { 5 }$ , b = -8, c = -5 
B)  $a = - \frac { 4 } { 5 }$ , b = 8, c = 5 
C)  $a = \frac { 4 } { 5 }$ , b = 8, c = 5 
D)  $a = \frac { 4 } { 5 }$ , b = -8, c = 5

Question 8

An open box is to be made from a square piece of cardboard, 54 inches by 54 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

36 inches

Question 9

Find the intervals of increase and decrease for the function $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }$ .

Accepted Answer

Decreasing for x < -

Question 10

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

Accepted Answer

A)  Increasing on x$\le$ -2 and x $\ge$  3, decreasing on - 2 $\le$ x$\le$ 3 
B)  Increasing on x  2, decreasing on -3  2 
D)  Increasing on x  3 
A)  Increasing on x$\le$ -2 and x $\ge$  3, decreasing on - 2 $\le$ x$\le$ 3 
B)  Increasing on x  2, decreasing on -3  2 
D)  Increasing on x  3

Question 11

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Accepted Answer

Concave up

Question 12

A commuter train carries 600 passengers each day from a suburb to a city. It now costs $1 per person to ride the train. A study shows that 50 additional people will ride the train for each 5 cent reduction in fare. What fare should be charged in order to maximize total revenue?

Accepted Answer

A)  78 cents 
B)  79 cents 
C)  80 cents 
D)  85 cents 
A)  78 cents 
B)  79 cents 
C)  80 cents 
D)  85 cents

Question 13

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$6.

Accepted Answer

The answer of Find the absolute minimum of the function...

Question 14

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 8$ on the interval -1 $\le$ x $\le$ 2.

Accepted Answer

Maximum is

Question 15

Determine where the graph of $g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }$ is concave up and concave down.

Accepted Answer

Concave up for x < -

Question 16

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)  $\left( - \sqrt { 3 } , - \frac { \sqrt { 3 } } { 4 } \right) , ( 0,0 ) , \left( \sqrt { 3 } , \frac { \sqrt { 3 } } { 4 } \right)$

Question 17

Which non-negative number exceeds its own 8th power by the greatest amount?

Accepted Answer

The answer of Which non-negative number exceeds its own 8th...

Question 18

Find two non-negative numbers whose sum is 8 and whose product is as large as possible.

Accepted Answer

A)  2 and 6 
B)  1 and 7 
C)  0 and 8 
D)  4 and 4 
A)  2 and 6 
B)  1 and 7 
C)  0 and 8 
D)  4 and 4

Question 19

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 9 x ^ { 4 } - 24 x ^ { 3 } + 18 x ^ { 2 } + 3$

Accepted Answer

(0, 3) rel

Question 20

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

Accepted Answer

A)  for x  1 
B)  for -1 < x < 1 
C)  Everywhere 
D)  Nowhere 
A)  for x  1 
B)  for -1 < x < 1 
C)  Everywhere 
D)  Nowhere

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

A Florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 500 oranges. The average yield will decrease by 5 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield?

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

Find all intervals where the derivative of the function shown below is negative.

Use the second derivative test to find the relative maxima and minima of the function $f ( x ) = 4 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1$ .

The revenue derived from the production of x units of a particular commodity is $R ( x ) = \frac { 48 x - x ^ { 2 } } { x ^ { 2 } + 48 }$ million dollars. What level of production results in maximum revenue? What is the maximum revenue? Round numbers to two decimal places, if necessary.

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).

An open box is to be made from a square piece of cardboard, 54 inches by 54 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?

Find the intervals of increase and decrease for the function $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }$ .

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

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

A commuter train carries 600 passengers each day from a suburb to a city. It now costs $1 per person to ride the train. A study shows that 50 additional people will ride the train for each 5 cent reduction in fare. What fare should be charged in order to maximize total revenue?

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 6.

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 8$ on the interval -1 $\le$ x $\le$ 2.

Determine where the graph of $g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }$ is concave up and concave down.

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

Which non-negative number exceeds its own 8th power by the greatest amount?

Find two non-negative numbers whose sum is 8 and whose product is as large as possible.

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 9 x ^ { 4 } - 24 x ^ { 3 } + 18 x ^ { 2 } + 3$

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

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

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Exam 3: Additional Applications of the Derivative

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

A Florida citrus grower estimates that if 60 orange trees are planted, the average yield per tree will be 500 oranges. The average yield will decrease by 5 oranges per tree for each additional tree planted on the same acreage. How many trees should the grower plant to maximize the total yield?

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

Find all intervals where the derivative of the function shown below is negative.

Use the second derivative test to find the relative maxima and minima of the function f(x)=4x3+12x2−36x+1f ( x ) = 4 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1f(x)=4x3+12x2−36x+1 .

Find constants a, b, and c so that the graph of the function f(x)=ax2+bx+cf ( x ) = a x ^ { 2 } + b x + cf(x)=ax2+bx+c has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).

An open box is to be made from a square piece of cardboard, 54 inches by 54 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?

Find the intervals of increase and decrease for the function f(x)=x−1x2+3f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }f(x)=x2+3x−1​ .

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

Determine where the graph of f(x)=x23f ( x ) = \frac { x ^ { 2 } } { 3 }f(x)=3x2​ is concave up and concave down.

A commuter train carries 600 passengers each day from a suburb to a city. It now costs $1 per person to ride the train. A study shows that 50 additional people will ride the train for each 5 cent reduction in fare. What fare should be charged in order to maximize total revenue?

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 ≤\le≤ x ≤\le≤ 6.

Find the absolute maximum and absolute minimum of f(x)=x4−8x2+8f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 8f(x)=x4−8x2+8 on the interval -1 ≤\le≤ x ≤\le≤ 2.

Determine where the graph of g(x)=1x2+3g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }g(x)=x2+31​ is concave up and concave down.

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

Which non-negative number exceeds its own 8th power by the greatest amount?

Find two non-negative numbers whose sum is 8 and whose product is as large as possible.

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. f(x)=9x4−24x3+18x2+3f ( x ) = 9 x ^ { 4 } - 24 x ^ { 3 } + 18 x ^ { 2 } + 3f(x)=9x4−24x3+18x2+3

Determine where the graph of f(x)=x4−6x2f ( x ) = x ^ { 4 } - 6 x ^ { 2 }f(x)=x4−6x2 is concave up.

Functions, Graphs, and Limits

Differentiation: Basic Concepts

Exponential and Logarithmic Functions

Integration

Additional Topics in Integration

Calculus of Several Variables

Appendix: Algebra Review

Filters

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

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 $f ( x ) = 4 x ^ { 3 } + 12 x ^ { 2 } - 36 x + 1$ .

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (5, 25) and crosses the y-axis at (0, 5).

Find the intervals of increase and decrease for the function $f ( x ) = \frac { x - 1 } { x ^ { 2 } + 3 }$ .

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

Determine where the graph of $f ( x ) = \frac { x ^ { 2 } } { 3 }$ is concave up and concave down.

Find the absolute minimum of the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 }$ on the interval -1 $\le$ x $\le$ 6.

Find the absolute maximum and absolute minimum of $f ( x ) = x ^ { 4 } - 8 x ^ { 2 } + 8$ on the interval -1 $\le$ x $\le$ 2.

Determine where the graph of $g ( x ) = \frac { 1 } { x ^ { 2 } + 3 }$ is concave up and concave down.

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

Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 9 x ^ { 4 } - 24 x ^ { 3 } + 18 x ^ { 2 } + 3$

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