Question 1

What value of x gives a maximum for   ? Restrict your attention to the horizontal span of 0 to 80.Round your answer to two decimal places. ​

Accepted Answer

A)  65.48 
B)  19.82 
C)  68.68 
D)  18.23 
A)  65.48 
B)  19.82 
C)  68.68 
D)  18.23 D

Question 2

One model for the growth rate G, in thousands per year, of a certain species depends on the population N, in thousands.The relationship is   ​ What population corresponds to the maximum growth rate, and what is that growth rate? Round your answers to two decimal places.
​

Accepted Answer

A)  The maximum growth rate of 1.75 thousand per year corresponds to a population of 2.50 thousand. 
B)  The maximum growth rate of 2.50 thousand per year corresponds to a population of 1.75 thousand. 
C)  The maximum growth rate of 4.40 thousand per year corresponds to a population of 4.24 thousand. 
D)  The maximum growth rate of 4.24 thousand per year corresponds to a population of 4.40 thousand. 
A)  The maximum growth rate of 1.75 thousand per year corresponds to a population of 2.50 thousand. 
B)  The maximum growth rate of 2.50 thousand per year corresponds to a population of 1.75 thousand. 
C)  The maximum growth rate of 4.40 thousand per year corresponds to a population of 4.24 thousand. 
D)  The maximum growth rate of 4.24 thousand per year corresponds to a population of 4.40 thousand. A

Question 3

For a certain epidemic the number N of new cases expected in a day depends on the number t of days since the outbreak began.The relationship is   . ​
What is the greatest number of new cases that occur on a day, and on what day does that happen? Round both answers to the nearest whole number.

Accepted Answer

A)  A maximum of 41 new cases occurs on day 24. 
B)  A maximum of 21 new cases occurs on day 38. 
C)  A maximum of 24 new cases occurs on day 41. 
D)  A maximum of 38 new cases occurs on day 21. 
A)  A maximum of 41 new cases occurs on day 24. 
B)  A maximum of 21 new cases occurs on day 38. 
C)  A maximum of 24 new cases occurs on day 41. 
D)  A maximum of 38 new cases occurs on day 21. D

Question 4

What is the minimum value of   ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places. ​

Accepted Answer

A)  -74.69 
B)  5.47 
C)  -70.70 
D)  4.44 
A)  -74.69 
B)  5.47 
C)  -70.70 
D)  4.44

Question 5

The number N, measured in millions of students, enrolled in high school t years after 1965 is given approximately by ​   .
According to this model, in what year (as a date) did high school enroll reach a maximum, and what was the maximum enrollment? Round your answers to two decimal places.
​

Accepted Answer

A)  The maximum enrollment of 13.68 million occurred in 1977. 
B)  The maximum enrollment of 12.00 million occurred in 1981.00. 
C)  The maximum enrollment of 13.44 million occurred in 1984. 
D)  There is no maximum value. 
A)  The maximum enrollment of 13.68 million occurred in 1977. 
B)  The maximum enrollment of 12.00 million occurred in 1981.00. 
C)  The maximum enrollment of 13.44 million occurred in 1984. 
D)  There is no maximum value.

Question 6

One model for expenditures at high schools gives the cost C, in dollars per pupil, as a function of the number n of students enrolled.The relationship is   . ​
Find the enrollment and cost that correspond to the minimum cost per pupil.Round your answers to the nearest whole number.

Accepted Answer

A)  An enrollment of 1909 gives a minimum cost of $340 per pupil. 
B)  An enrollment of 340 gives a minimum cost of $1909 per pupil. 
C)  An enrollment of 1911 gives a minimum cost of $344 per pupil. 
D)  An enrollment of 344 gives a minimum cost of $1911 per pupil. 
A)  An enrollment of 1909 gives a minimum cost of $340 per pupil. 
B)  An enrollment of 340 gives a minimum cost of $1909 per pupil. 
C)  An enrollment of 1911 gives a minimum cost of $344 per pupil. 
D)  An enrollment of 344 gives a minimum cost of $1911 per pupil.

Question 7

What is the minimum value of   ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

Accepted Answer

A)  2.09 
B)  10.03 
C)  0.75 
D)  13.45 
A)  2.09 
B)  10.03 
C)  0.75 
D)  13.45

Question 8

What value of x gives the minimum value of   ? Restrict your attention to the horizontal span of 0 to 19.Round your answer to two decimal places. ​

Accepted Answer

A)  -3.65 
B)  3.67 
C)  -0.17 
D)  2.53 
A)  -3.65 
B)  3.67 
C)  -0.17 
D)  2.53

Question 9

What is the minimum value for   ? Restrict your attention to the horizontal span of 40 to 46.Be careful.You are asked about a minimum, not a maximum.Round your answer to two decimal places. ​

Accepted Answer

A)  179.02 
B)  46.00 
C)  175.66 
D)  42.63 
A)  179.02 
B)  46.00 
C)  175.66 
D)  42.63

Question 10

For which value of x does   reach a maximum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

Accepted Answer

A)  1.41 
B)  15.89 
C)  0.36 
D)  19.55 
A)  1.41 
B)  15.89 
C)  0.36 
D)  19.55

Question 11

No function can have both a maximum and a minimum value.

Accepted Answer

A) True 
 B)False

Question 12

Graphs can be used to find maxima and minima for functions.

Accepted Answer

A) True 
 B)False

Question 13

What is the minimum value of   ? Restrict your attention to the horizontal span of 0 to 1.Round your answer to two decimal places. ​

Accepted Answer

A)  0.13 
B)  1.48 
C)  2.06 
D)  5.46 
A)  0.13 
B)  1.48 
C)  2.06 
D)  5.46

Question 14

For which value of x does   reach a maximum? Restrict your attention to the horizontal span of 0 to 12.Round your answer to two decimal places. ​

Accepted Answer

A)  1.55 
B)  3.17 
C)  0.06 
D)  7.09 
A)  1.55 
B)  3.17 
C)  0.06 
D)  7.09

Question 15

A can is made from 6 square inches of aluminum.We can calculate both the height   , in inches, and the volume V, in cubic inches, from the radius r, in inches, of the can:   ​ Find the radius and height of the can of maximum volume.Round your answers to two decimal places.
​

Accepted Answer

A)  The radius is 1.13 inches, and the height is 0.56 inches. 
B)  The radius is 3.34 inches, and the height is 1.83 inches. 
C)  The radius is 1.83 inches, and the height is 3.34 inches. 
D)  The radius is 0.56 inches, and the height is 1.13 inches. 
A)  The radius is 1.13 inches, and the height is 0.56 inches. 
B)  The radius is 3.34 inches, and the height is 1.83 inches. 
C)  The radius is 1.83 inches, and the height is 3.34 inches. 
D)  The radius is 0.56 inches, and the height is 1.13 inches.

Question 16

For which value of x does   reach a minimum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

Accepted Answer

A)  13 
B)  11.44 
C)  0.63 
D)  0 
A)  13 
B)  11.44 
C)  0.63 
D)  0

Question 17

Maxima for functions can be found at peaks or at endpoints.

Accepted Answer

A) True 
 B)False

Question 18

Minima always occur at endpoints.

Accepted Answer

A) True 
 B)False

Question 19

Find the maximum value of   .Restrict your attention to the horizontal span of 0 to 9.Round your answer to two decimal places. ​

Accepted Answer

A)  1.13 
B)  20.06 
C)  2.72 
D)  23.89 
A)  1.13 
B)  20.06 
C)  2.72 
D)  23.89

Question 20

What value of x gives the minimum value of   ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

Accepted Answer

A)  29.38 
B)  5.39 
C)  4.05 
D)  33.06 
A)  29.38 
B)  5.39 
C)  4.05 
D)  33.06

Exam 10: Optimization

What value of x gives a maximum for ? Restrict your attention to the horizontal span of 0 to 80.Round your answer to two decimal places. ​

One model for the growth rate G, in thousands per year, of a certain species depends on the population N, in thousands.The relationship is ​ What population corresponds to the maximum growth rate, and what is that growth rate? Round your answers to two decimal places. ​

For a certain epidemic the number N of new cases expected in a day depends on the number t of days since the outbreak began.The relationship is . ​ What is the greatest number of new cases that occur on a day, and on what day does that happen? Round both answers to the nearest whole number.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places. ​

The number N, measured in millions of students, enrolled in high school t years after 1965 is given approximately by ​ . According to this model, in what year (as a date) did high school enroll reach a maximum, and what was the maximum enrollment? Round your answers to two decimal places. ​

One model for expenditures at high schools gives the cost C, in dollars per pupil, as a function of the number n of students enrolled.The relationship is . ​ Find the enrollment and cost that correspond to the minimum cost per pupil.Round your answers to the nearest whole number.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 19.Round your answer to two decimal places. ​

What is the minimum value for ? Restrict your attention to the horizontal span of 40 to 46.Be careful.You are asked about a minimum, not a maximum.Round your answer to two decimal places. ​

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

No function can have both a maximum and a minimum value.

Graphs can be used to find maxima and minima for functions.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 1.Round your answer to two decimal places. ​

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 12.Round your answer to two decimal places. ​

A can is made from 6 square inches of aluminum.We can calculate both the height , in inches, and the volume V, in cubic inches, from the radius r, in inches, of the can: ​ Find the radius and height of the can of maximum volume.Round your answers to two decimal places. ​

For which value of x does reach a minimum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

Maxima for functions can be found at peaks or at endpoints.

Minima always occur at endpoints.

Find the maximum value of .Restrict your attention to the horizontal span of 0 to 9.Round your answer to two decimal places. ​

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

Functions Given by Formulas

Functions Given by TableChas

Functions Given by Graphs

Functions Given by Words

Table and Trends

Graphs

Solving Linear Equations

Solving Nonlinear Equations

Inequalities

The Geometry of Lines

Linear Functions

Modeling Data with Linear Functions

Linear Regression

Systems of Equations

Exponential Growth and Decay

Constant Percentage Change

Modeling Exponential Data

Modeling Nearly Exponential Data

Logarithmic Functions

Logistic Functions

Power Functions

Modeling Data with Power Functions

Combining and Decomposing Functions

Quadratic Functions

Higher Degree Polynomials and Rational Functions

Velocity

Rates of Change for Other Functions

Estimating Rates of Change

Equations of Change Linear and Exponential Functions

Equations of Change Graphical Solutions

Filters

What value of x gives a maximum for ? Restrict your attention to the horizontal span of 0 to 80.Round your answer to two decimal places.

One model for the growth rate G, in thousands per year, of a certain species depends on the population N, in thousands.The relationship is What population corresponds to the maximum growth rate, and what is that growth rate? Round your answers to two decimal places.

For a certain epidemic the number N of new cases expected in a day depends on the number t of days since the outbreak began.The relationship is . What is the greatest number of new cases that occur on a day, and on what day does that happen? Round both answers to the nearest whole number.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places.

The number N, measured in millions of students, enrolled in high school t years after 1965 is given approximately by . According to this model, in what year (as a date) did high school enroll reach a maximum, and what was the maximum enrollment? Round your answers to two decimal places.

One model for expenditures at high schools gives the cost C, in dollars per pupil, as a function of the number n of students enrolled.The relationship is . Find the enrollment and cost that correspond to the minimum cost per pupil.Round your answers to the nearest whole number.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places.

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 19.Round your answer to two decimal places.

What is the minimum value for ? Restrict your attention to the horizontal span of 40 to 46.Be careful.You are asked about a minimum, not a maximum.Round your answer to two decimal places.

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places.

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 1.Round your answer to two decimal places.

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 12.Round your answer to two decimal places.

A can is made from 6 square inches of aluminum.We can calculate both the height , in inches, and the volume V, in cubic inches, from the radius r, in inches, of the can: Find the radius and height of the can of maximum volume.Round your answers to two decimal places.

For which value of x does reach a minimum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places.

Find the maximum value of .Restrict your attention to the horizontal span of 0 to 9.Round your answer to two decimal places.

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places.