The following graph shows the number, in millions, of students graduating from high school in the U.S.

The following graph shows the number, in millions, of students graduating from high school in the U.S.

The following graph shows the number, in millions, of students graduating from high school in the U.S. 150 150 Peter


1. The following graph shows the number, in millions, of students graduating from high school in the U.S., in the given year.

a. Explain in practical terms what N(1989) means and find its value.

b. Use functional notation to express the number of graduates in 1988 and estimate its value.

c. Find the average rate of change per year during the period 1989 – 1991.

d. Use part c to try to estimate N(1992)

0. The following graph shows a population N=N(d), where d is the date.

a. Is the function increasing, decreasing, or neither for d from 2006 to 2016?

b. When is the function concave down and when is it concave up? Give the ranges in years.

c. At what date is there an inflection point?

0. You currently have $780 in your bank account, which pays no interest. You withdraw $39 each week. Find a formula for the balance B, in dollars, in the account after t weeks.

0. One cell phone plan charges a flat monthly rate of $39.95 with extra charges of $0.10 per text message after the first 100 text messages.

a. Choose letters to represent the variables

b. Write a formula to express the cell phone charges as a function of the number of text messages (assume that the number of text messages is at least 100.)

c. Use functional notation to represent the cost of the cell phone if you have 450 text messages.

d. Use the formula from part b to calculate the cost.

e. Write a formula to express the charges assuming the number of text messages is less than 100.

0. It starts to snow when there is already snow on the ground. The depth, in inches, of the snow t hours later is given by  S(t)=10.00 + 0.4t.

a. How much snow was on the ground when the snow started to fall?

b. By how much did the depth of snow increase from hour 3.00 to hour 6.00?

c. Snow ceases to fall after 10 hours. What is the resulting depth of snow on the ground?

0. The following table shows the U.S. population, in millions, in the given year.

d = year






N= population in millions






a. Calculate the average rate of change from 1980 to 1990

b. Use the average rate of change to estimate the U.S. population in 1982.

0. A hot potato is placed on the kitchen counter to cool. Its temperature after t minutes is given by T(t). The temperature of the kitchen is 74 degrees Fahrenheit. What is the limiting value of ​T?

0. The graph below shows the growth rate G, in water fleas per day, of a population of water fleas as a function of the population size N.  Calculate the average rate of change in G from 130 to 160 water fleas. Round to 3 decimal places.

0. You sell earrings for $6.33 each. You paid $57.21 for supplies to make the earrings.  Write a formula for the profit P, in dollars, you make from making and selling E earrings, where the most earrings you can make with the supplies is 26.

0. The weight W, in pounds, of a rock is proportional to its volume V, in cubic inches.  The constant of proportionality is the density, d, in pounds per cubic inch.

a. Write a formula that expresses the proportionality relationship.

b. What is the weight of a rock that has a density of 1.02 pounds per cubic inch and a volume of 14 cubic inches?

0. The world record time for a certain track event in 2000 was 74.98 seconds.  In the ensuing years, the record time has decreased by 0.94 seconds each year.  

a. Write a formula for the record R, in seconds, t years after 2000.

b. Use functional notation to express the record in 2002.

c. Calculate the value you found in part B

0. You start with $32.28 in your piggy bank and add $9.28 each week.  Write a formula for the amount P of money, in dollars, in your piggy bank after t weeks.

0. The graph below shows the value, in dollars, of a foreign currency t years after 2000.

In what year from 2000 to 2020 did the value of the foreign currency reach its maximum, and what was that maximum value? 


0. The number of deer on the George Reserve t years after introduction is given by

N=6.210.035 + 0.45t   deer.

a. How many deer were introduced into the deer reserve?

b. Calculate N(4) and explain the meaning of the number you have calculated.

c. Find the carrying capacity for deer in the reserve.

d. Explain how the population varies with time. Include the average rate of increase over each 2-year period for the first 8 years.

0. A gliding bird’s movement is measured in terms of the rate at which the bird’s altitude decreases, the sinking speed s, and the airspeed u. For a gliding pigeon, these are related by

                             s=u32500+ 25u .

Here s and u are measured in meters per second.

a. Make a graph of the sinking speed s as a function of the airspeed u for values of u up to 20 meters per second.

b. At what airspeed does the smallest sinking speed occur?

c. If the airspeed is very slow, does the pigeon sink quickly or slowly? 

0. The profit P, in millions of dollars, that a manufacturer makes is a function of the number N, in millions, of items produced in a year, and the formula is 

                           P=10N-N2-6.34  .

A negative quantity for P represents a negative profit—that is, a loss—and the formula is valid up to a level of 10 million items produced.

a. Express using functional notation the profit at a production level of 7 million items.

b. What is the loss at a production level of N = 0?

c. Determine the two break-even points for this manufacturer—that is, the two production levels at which the profit is zero.

d. Determine the production level that gives maximum profit, and determine the amount of the maximum profit.

0. If a completely full 5-gallon water jug is drained through a spigot, the depth is given by the formula 

D = 0.265 t2 – 4.055t +15.5

where t is the time, in minutes, that the spigot is open and D is the depth, in inches, above the spigot.

a. Make a graph of D versus t. Include values of t up to 7.5 minutes.

b. Calculate D(3) and explain in practical terms what your answer means.

c. When will the water jug be completely drained?

d. Does the water drain faster near the beginning or when the water is mostly drained?

0. We want to form a free-standing rectangular pen. Let W be the width, in feet, and L the length, in feet. Let F be the total amount of fence needed. Assume that the total area of the pen will be 144 square feet.

a. Write a formula for F in terms of W and L.

b. Express, as an equation involving W and L, the requirement that the total area of the pen be 144 square feet.

c. Solve the equation you found in part b for W.

d. Use your answers to parts a and c to find a formula for F in terms of L alone.

e. Make a graph of F versus L for values of L up to 24 feet.

f. Determine the dimensions of the pen that require the minimum amount of fence. Explain how your graph from part e illustrates or confirms your answer.

0. The length of North Sea sole, a species of fish, can be determined using the von Bertalanffy model by the formula


Here t is age, in years, and L is length, in inches.

a. Make a graph of L versus t. Include values of t from 1 to 10 years.

b. Use functional notation to express the length of a 4-year-old sole, and then calculate that value.

c. At what age will a sole be 10 inches long?

d. What is the limiting length for a North Sea sole?

0. The three principal measures of temperature are Fahrenheit F, Celsius C, and Kelvin K. These are related by the formulas below:

F = 1.8C + 32

K = C + 273.15

a. Solve each of the equations for C.

b. Find a formula expressing F in terms of K.

c. Find a formula expressing K in terms of F.

d. Express the temperature 0 degrees Celsius in terms of degrees Fahrenheit and in terms of kelvins (degrees on the Kelvin scale).

e. Express the temperature 72 degrees Fahrenheit in terms of degrees Celsius and in terms of kelvins.

0. The red curve represents the temperature of a metal bar in degrees Fahrenheit as a function of time, t, in minutes. 

a. Write the inequality for the time when the bar is at least 720 degrees.

b. Write the inequality for the time when the bar is at least 700 degrees.

0. The cost function, C(n) = 743 – 0.36n + 0.00012n2 is graphed below

a. At what value of n does the minimum cost occur?

b. What is the minimum?