Chapter 1

Let the distance in feet that a car travels in \(t\) seconds be given by \(d(t)=8 t^{2}\) for \(0 \leq t \leq 6\) (a) Find \(d(t+h)\) (b) Find the difference quotient for \(d\) and simplify. (c) Evaluate the difference quotient when \(t=4\) and \(h=0.05 .\) Interpret your result.

Determine if \(y\) is a function of \(x\). $$ x=y^{4} $$

Predict the number of tick marks on the positive \(x\) -axis and the positive y-axis. Then show the viewing rectangle on your graphing calculator. $$ [1980,1995,1] \text { by }[12000,16000,1000] $$

Let the number of gallons \(G\) of water in a pool after \(t\) hours be given by \(G(t)=4000-100 t\) for \(0 \leq t \leq 40\) (a) Find \(G(t+h)\) (b) Find the difference quotient. Interpret your result.

Determine if \(y\) is a function of \(x\). $$ y^{2}=x+1 $$

Movement of the Pacific Plate The Pacific plate (the floor of the Pacific Ocean) near Hawaii is moving at about 0.000071 kilometer per year. This is about the speed at which a fingernail grows. Use scientific notation to determine how many kilometers the Pacific plate travels in one million years.

Predict the number of tick marks on the positive \(x\) -axis and the positive y-axis. Then show the viewing rectangle on your graphing calculator. $$ [1800,2000,20] \text { by }[5,20,5] $$

What does the average rate of change represent for a linear function? What does it represent for a nonlinear function? Explain your answers.

Determine if \(y\) is a function of \(x\). $$ \sqrt{x+1}=y $$

What is the formula for the difference quotient? Given a formula for \(f(x),\) explain how to find \(f(x+h) .\) Give an example.

The Pacific plate (the floor of the Pacific Ocean) near Hawaii is moving at about 0.000071 kilometer per year. This is about the speed at which a fingernail grows. Use scientific notation to determine how many kilometers the Pacific plate travels in one million years.

Suppose that a function \(f\) has a positive average rate of change from 1 to \(4 .\) Is it correct to assume that function \(f\) only increases on the interval \([1,4] ?\) Make a sketch to support your answer.

The amount of federal debt changed dramatically during the 30 years from 1970 to 2000 . (Sources: Department of the Treasury, Bureau of the Census.) A. In 1970 the population of the United States was \(203,000,000\) and the federal debt was \(\$ 370\) billion. Find the debt per person. B. In 2000 the population of the United States was approximately \(281,000,000\) and the federal debt was \(\$ 5.54\) trillion. Find the debt per person.

Determine if \(y\) is a function of \(x\). $$ x^{2}+y^{2}=70 $$

If \(f(x)=a x+b,\) what does the difference quotient for function \(f\) equal? Explain your reasoning.

Discharge of Water The Amazon River discharges water into the Atlantic Occan at an average rate of \(4,200,000\) cubic feet per second, the highest rate of any river in the world. Is this more or less than 1 cubic mile of water per day? Explain your calculations. (Source: The Guinness Book of Reconds I993.) (PICTURE NOT COPY)

Determine if \(y\) is a function of \(x\). $$ (x-1)^{2}+y^{2}=1 $$

Thickness of an Oil Film (Refer to Example 9.) A drop of oil measuring 0.12 cubic centimeter is spilled onto a lake. The oil spreads out in a circular shape having a diameter of 23 centimeters. Approximate the thickness of the oil film.

Determine if \(y\) is a function of \(x\). $$ x+y=2 $$

Make a scatterplot of the relation. $$ \\{(1,3),(-2,2),(-4,1),(-2,-4),(0,2)\\} $$

Thickness of Gold Foil (Refer to Example 9.) A flat, rectangular sheet of gold foil measures 20 centimeters by 30 centimeters and has a mass of 23.16 grams. If 1 cubic centimeter of gold has a mass of 19.3 grams, find the thickness of the gold foil. (Source: U. HaberSchaim, Introductory Physical Science.)

Determine if \(y\) is a function of \(x\). $$ y=|x| $$

Make a scatterplot of the relation. $$ \\{(6,8),(-4,-10),(-2,-6),(2,-5)\\} $$

Analyzing Debt A 1-inch-high stack of \(\$ 100\) bills contains about 250 bills. In 2000 the federal debt was approximately 5.54 trillion dollars. A. If the entire federal debt were converted into a stack of \(\$ 100\) bills, how many feet high would it be? B. The distance between Los Angeles and New York is approximately 2500 miles. Could this stack of \(\$ 100\) bills reach between these two cities?

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of inches in \(x\) feet

Make a scatterplot of the relation. $$ \\{(10,-20),(-40,50),(30,60),(-50,-80),(70,0)\\} $$

The volume \(V\) of a cone is given by \(V=\frac{1}{3} \pi r^{2} h,\) where \(r\) is its radius and \(h\) is its height. Find \(V\) when \(r=4\) inches and \(h=1\) foot. Round your answer to the nearest hundredth. (PICTURE NOT COPY)

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of quarts in \(x\) gallons

Make a scatterplot of the relation. $$ \\{(-1.2,0.6),(1.0,-0.5),(0.4,0.2),(-2.8,1.4)\\} $$

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of dollars in \(x\) quarters

The volume of a sphere is given by \(V=\frac{4}{3} \pi r^{3},\) where \(r\) is the radius of the sphere. Calculate the volume if the radius is 3 feet. Approximate your answer to the nearest tenth.

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of quarters in \(x\) dollars

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of seconds in \(x\) days

Write a symbolic representation (formula) for a function \(g\) that calculates the given quantify. Then evaluate \(g(10)\) and interpret the result. The number of feet in \(x\) miles

Give an example of a relation that has meaning in the real world. Give an example of an ordered pair ( \(x, y\) ) that is in your relation. Does the ordered pair \((y, x)\) also have meaning? Explain your answers.

Writing about Mathematics Describe some basic sets of numbers that are used in mathematics.

Do the mean and median represent the same thing? Explain your answer and give an example.

Writing about Mathematics Suppose that a positive number \(a\) is written in scientific notation as \(a=b \times 10^{n}\), where \(n\) is an integer and \(1 \leq b \leq 10 .\) Explain what \(n\) indicates about the size of \(a\)

Going Green The average person uses 2200 paper napkins in one year. Write the formula for a function \(N\) that calculates the number of paper napkins that the average person uses in \(x\) years. Evaluate \(N(3)\) and interpret your result.

Going Green The average top-loading washing machine uses about 40 gallons of water per load of clothes. Write the formula for a function \(W\) that calculates the number of gallons of water used while washing \(x\) loads of clothes. Evaluate \(W(30)\) and interpret your result.

Air Temperature (Refer to Example 6 .) When the relative humidity is \(100 \%,\) air cools \(5.8^{\circ} \mathrm{F}\) for every 1 -mile increase in altitude. Give verbal, symbolic, graphical, and numerical representations of a function \(f\) that computes this change in temperature for an increase in altitude of \(x\) miles for \(0 \leq x \leq 3\).

Distance to Lightning Find a formula for a function \(f\) that computes the distance between an observer and a lightning bolt when the speed of sound is 1150 feet per second. Evaluate \(f(15)\) and interpret the result.

Explain how you could use a complete numerical representation (table) for a function to determine its domain and range.

Explain in your own words what a function is. How is a function different from a relation?