Chapter 2

The average price of an 8 -ounce container of yogurt in upstate New York was 35 cents in \(1970 .\) In 2000 the average price had risen to 89 cents. (a) What is the price increase? (b) What is the percent increase in price? (c) What is the average rate of change in price from 1970 to \(2000 ?\)

For Problems 1 through 7, give exact answers, not numerical approximations. Find the radius of the circle whose area is 2 square inches.

This problem applies the definitions of proportionality (also referred to as direct proportionality), and inverse proportionality. \(\begin{array}{lll}2.2 & \text { A Pocketful of Functions: Some Basic Examples } & \end{array}\) If \(P\) is the pressure of a gas, \(T\) is its temperature, and \(V\) is its volume, then the combined gas law tells us that for a fixed mass of gas $$ \frac{P V}{T}=\text { a constant. } $$ From this statement, demonstrate the following. (a) If temperature is held constant, then pressure is inversely proportional to volume (Boyle's Law). (This should make sense intuitively; as pressure increases, volume decreases.) (b) If pressure is held constant, then volume is directly proportional to temperature (Law of Charles and Gay-Lussac). (This too should make sense intuitively; as the temperature of a gas goes up, volume goes up as well.)

For Problems 1 through 7, give exact answers, not numerical approximations. Find the diameter of the circle whose circumference is 7 inches.

Physicists define the work done by a force on an object to be the magnitude of the force, \(F\), times the distance, \(d\), that the object is moved. Notice that this definition is different from our conversational use of the word "work." For instance, if you stand holding a 50 -pound object stationary for an hour, then according to the physicists' definition you have done no work because the object has not moved. According to the physicists' definition, if force is kept constant is the work done proportional to the distance the object moves. Explain.

A function \(f\) is decreasing throughout its domain \([-8,-1]\). Can we determine where \(f\) takes on its largest value? Does your answer depend upon whether or not \(f\) is continuous?

Suppose \(f\) is a continuous function with domain \([-4,7] ; f\) is decreasing on \([-4,4]\) and increasing on \([4,7]\). (a) Can you determine where \(f\) takes on its lowest value? If this can be done, do it. If not, can you narrow down the selection to a few possibilities? Can you determine the lowest value of \(f ?\) (b) Can you determine where \(f\) takes on its highest value? If this can be done, do it. If not, can you narrow down the selection to a few possibilities? Can you determine the highest value of \(f ?\)

For Problems 1 through 7, give exact answers, not numerical approximations. How long is the diagonal of a square whose sides are 5 inches?

Consider the statement "Colleges report that their tuition increases are slowing down." Suppose we set \(t=0\) to be three years before this statement was made and measure time in years. If we let \(T(t)\) be the average college tuition in year \(t\), put the following expressions in ascending order (the smallest first), assuming the statement is true. $$ T(3)-T(2) \quad 0 \quad T(2)-T(1) \quad \frac{T(3)-T(1)}{3-1} $$

For Problems 1 through 7, give exact answers, not numerical approximations. A rectangle is 3 meters long and 2 meters high. How long is the diagonal?

Use the geometric interpretation of absolute value to solve the following equations and inequalities. Display the solutions on a number line. (a) \(|x+3|<2\) (b) \(|x-5| \leq 3\) (c) \(|x-a|=b, \quad\) where \(a\) and \(b\) are positive (d) \(|x+a| \leq a, \quad\) where \(a\) is positive

The Boston Globe (August 31,1999, p. 1 ) reports: "While the number of AIDS deaths continues to drop nationally, the rapid rate of decline that had been attributed to new drugs is starting to slow dramatically." The newspaper supplies the following data. $$ \begin{array}{ll} \hline \text { Year } & \text { Number of Deaths Attributed to AIDS } \\ \hline 1995 & 149,351 \\ 1996 & 36,792 \\ 1997 & 21,222 \\ 1998 & 17,047 \end{array} $$ Let \(D(t)\) be the number of deaths from AIDS in year \(t\), where \(t\) is measured in years and \(t=0\) corresponds to 1995 . (a) Is \(D(t)\) positive or negative? Increasing or decreasing? (b) What is the average rate of change of \(D(t)\) from \(t=0\) to \(t=1 ?\) What is the percent change in \(D(t)\) over that year? (c) What is the average rate of change of \(D(t)\) from \(t=1\) to \(t=2 ?\) What is the percent change in \(D(t)\) over that year? (d) What is the average rate of change of \(D(t)\) from \(t=2\) to \(t=3 ?\) What is the percent change in \(D(t)\) over that year?

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \(x^{2}+1=6\)

Use the algebraic interpretation of absolute value to solve each of the following. Please display your answers on a number line. (a) \(|2 x+1 / 2| \leq 6\) (b) \(|3 x-4|>8\)

In the peak of apple season in rural Vermont, migratory workers are often hired to pick apples. In this problem, model the situation in one orchard. Depending upon your interpretation of the situation, your model may differ from those of your classmates. Discuss your answers with others. (a) Sketch a graph of the time it takes to harvest the crop of apples as a function of the number of people picking apples. Label your axes. (b) Is the graph you drew a straight line? Why or why not? (c) Is the graph you drew continuous or discontinuous? Explain. (d) Does the graph you drew intersect the vertical axis? The horizontal axis? If so, where? If not, why not?

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \((\pi x)^{2}=\pi x\). (There are two answers.)

Solve the following inequalities. Display your answers on a number line using interval notation. (a) \(-2 x-7<-8\) (b) \(|-2 x-8| \geq 2\) (c) \(|-2 x-8|<2\)

Sketch a graph of a continuous function defined for all real numbers with all three characteristics listed. If it is impossible to do this, say so, and draw the graph of a function with the three characteristics on the domain \([-1,1]\). (a) \(f\) is positive, increasing, and concave up (b) \(f\) is positive, increasing, and concave down (c) \(f\) is negative, increasing, and concave up (d) \(f\) is negative, increasing, and concave down (e) \(f\) is positive, decreasing, and concave up (f) \(f\) is positive, decreasing, and concave down (g) \(f\) is negative, decreasing, and concave up (h) \(f\) is negative, decreasing, and concave down

The velocity of an object is given in miles per hour by \(v(t)=2 t^{5}-6 t^{3}+2 t^{2}+1\) over the time interval \(-2 \leq t \leq 2\), where \(t\) is measured in hours. Use your graphing calculator to answer the following questions. (a) Sketch a graph of the velocity function over the time interval \(-2 \leq t \leq 2\). (b) Approximately when does the object change direction? Please give answers that are off by no more than \(0.05 .\) (Either use the "zoom" feature of your calculator or change the domain until you can answer this question. If your calculator has an equation solver, use that as well and compare the answers you arrive at graphically with the answers you get using the equation solver.) (c) On the interval \(-2 \leq t \leq 2\), approximately when is the object going the fastest? How fast is it going at that time? (Give your answer accurate to within 0.1.) (d) When on the interval \(0 \leq t \leq 2\) is the velocity most negative? (Give an answer accurate to within 0.1.) When you zoom in on the graph here, what do you observe?

Let \(h(t)\) denote the height of a rocketship \(t\) seconds after takeoff. (a) Express the average rate of change of height of the rocket betweeen 2 and \(2.01\) seconds after takeoff in terms of the function \(h\). (b) Express the average rate of change of height (average vertical velocity) of the rocket on the time interval \([a, a+0.001]\) in terms of \(h\). (c) Express the average vertical velocity of the rocket on the time interval \([a, a+k]\).

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \(\pi^{2} x^{3}=\pi x^{2}\)

Solve these inequalities and explain your answers: Think carefully. (a) \(|3 x-4|>-4\) (b) \(|3 x-4|>0\) (c) \(|3 x-4|<-4\)

The displacement of an object is given by \(d(t)=2 t^{5}-6 t^{3}+2 t^{2}+1\) miles over the time interval \(-2 \leq t \leq 2\) where \(t\) is measured in hours. (a) Approximately when does the object change direction? Please give answers accurate to within \(0.1\). When you zoom in on the graph here, what do you observe? (b) Approximately when is the object's velocity positive? Negative? (c) Approximate the object's velocity at time \(t=0\).

(a) How many rational numbers are in the interval \([2,2.001] ?\) (b) How many irrational numbers are in the interval \([2,2.001] ?\)

Which of the following statements are true and which are not always true? For a statement to be true, it must always be true. If a statement is not always true, give a counterexample. To give a counterexample is to give an example of values for \(x\) and \(y\) for which the statement is false. (a) \(\left|x^{2}\right|=|x|^{2}\) (b) \(|x|=|-x|\) (c) \(|x-y|=|x|-|y|\) (d) \(|x+y|=|x|+|y|\) (e) \(\frac{|x|}{|y|}=\left|\frac{x}{y}\right|\) for \(y \neq 0\) (f) \(|x||y|=|x y|\) (Four of the statements are true.)

The number \(\pi\) lies between \(3.141592653489\) and \(3.141592653490\). How many other irrational numbers lie between these two?

Use absolute values to write the following statement more compactly: Whenever \(x\) is within \(0.02\) of \(7, f(x)\) differs from 19 by no more than \(0.3\).

Find the average rate of change of \(f(x)=x^{2}\) over each of the following intervals. (a) \([0,3]\) (b) \([1,4]\) (c) \([2,5]\) (d) \([a, a+3]\) (e) \([a, a+h]\)

Which of the following functions is continuous at \(x=2\) ? (a) \(f(x)=x+3\) (b) \(f(x)=\frac{x+3}{x-2}\) (c) \(f(x)=\frac{x^{2}+x-6}{x-2}\)

Before restrictions were placed on the distance that a backstroker could travel underwater in a race, Harvard swimmer David Berkoff set an American record for the event by employing the following strategy. In a 100-meter race in a 50-meter pool, Berkoff would swim most of the first 50 meters underwater (where the drag effect of turbulence was lower) then come up for air and swim on the water's surface (at a slightly lower speed) until the turn. He would then use a similar approach to the second 50 meters, but could not stay underwater as long due to the cumulative oxygen deprivation caused by the time underwater. Assume that Berkoff is swimming a 100 -meter race in Harvard's Blodgett pool (which runs 50 meters east to west). He starts on the east end, makes the 50-meter turn at the west end, and finishes the race at the east end. Sketch a graph of his velocity, taking east-to-west travel to have a positive velocity and west-to-east a negative velocity.

Find the average rate of change of \(g(t)=\frac{t}{t^{2}+2}+3 t\) over the intervals \([-1,1],[0,2]\), \([1,1+p]\)

(a) Is it possible for the graph of a function \(f\) with domain \([0,2]\) to have at most finitely many points with an irrational coordinate? If so, give such a function. (b) Is it possible for the graph of a function \(g\) with domain \(\\{0,1,2, \ldots\\}\) to have no points with an irrational coordinate? If so, give an example of such a function.

Let \(h(x)=|x| .\) Solve the following. Do parts (a) and (b) twice - once using an analytic approach and once using a geometric approach. (a) \(h(x+2) \leq 3\) (b) \(h(x-1)=5\) (c) \(h(x+3) \geq 0.1\) (d) \(h(3 x+1)>4\)

Which of the following functions is continuous at \(x=1 ?\) (a) \(f(x)=5\) (b) \(f(x)=\frac{5 x}{x}\) (c) \(f(x)=\frac{5 x(x-1)}{x-1}\) (d) \(f(x)=\frac{x^{2}(x-1)}{x-1}\)

A bicyclist does a one-mile climb at a constant speed of 12 miles per hour followed by a one-mile descent at a constant speed of 30 miles per hour. (a) Sketch a graph of distance traveled as a function of time. Assume the cyclist starts at \(t=0\) minutes, and be sure to label the times at which he reaches the top and bottom of the hill. (b) What is his average speed for the two miles? Is this the same as the average of 12 mph and \(30 \mathrm{mph}\) ? Explain why or why not.

Let \(h(x)=|x| .\) Solve the following. (a) \(2 h(x)>4\) (b) \(h(2 x-1) \leq 3\) (c) \(h\left(x^{2}-1\right) \geq 0\)

A backyard pool is a cylinder sitting above the ground and measuring \(3.5\) feet in height and 20 feet in diameter. (a) Express the volume of water in the pool as a function of the height \(h\) of the water. (Note: The domain of this function, the set of all acceptable inputs, is \(0 \leq h \leq 3.5 .\) ) (b) Sketch a graph of volume versus height. (c) What is the range of the function? Make sure this is indicated on your graph. (d) How much additional water is needed to increase the depth of water in the pool by 1 foot? By \(1 / 2\) foot? Is \(\frac{\Delta V}{\Delta h}\) constant? If so, what is it? (e) You've expressed the rate of change of volume with respect to height in terms of \(\mathrm{ft}^{3} / \mathrm{ft}\), but the volume of water is more likely to be measured in gallons or liters. Knowing that 1 gallon \(\approx 0.16054 \mathrm{ft}^{3}\), convert your answer to gallons/ft. (f) Is the volume of water in the pool directly proportional to the height of water?

In Problems 13 through 18, determine whether the function is even, odd, or neither. (a) \(f(x)=x^{2}+3 x^{4}\) (b) \(g(x)=\frac{1}{x^{2}+3 x^{4}}\)

During the 1996 Summer Olympics in Atlanta, track and field world records were set in both the men's 100 meters dash and the men's 200 meters dash. Donovan Bailey won the 100 in \(9.86\) seconds, and Michael Johnson won the 200 in \(19.32\) seconds. (a) What was the average speed of each runner? Which race had the higher average speed? Explain why you think this might be so. Please use graphs to illustrate your answer. (b) In 1996 the record for 400 meters was \(43.29\), set by Butch Reynolds. How does this average speed compare to the two given above? Does the "pattern" of the longer race having a higher average speed continue if we include the 400 meter? Give a possible reason for this.

(a) \(f(x)=2 x^{3}+3 x\) (b) \(g(x)=2 x^{3}+3 x+1\)

\(A\) and \(B\) are points on the graph of \(f(x)=\frac{1}{x^{2}}\). The \(x\) -coordinate of point \(A\) is 3 and the \(x\) -coordinate of point \(B\) is \((3+h) .\) Which of the expressions below correspond to the average rate of change of \(f\) on the interval \([3,3+h] ?\) (b) \(\frac{\frac{1}{3+h)^{2}}-\frac{1}{9}}{h}\) (c) \(\frac{\frac{1+h}{9}-\frac{1}{5}}{3}\) (d) \(\frac{1}{9}+\frac{1}{h}-\frac{1}{9}\) (e) \(\frac{\frac{1}{(3+h)^{2}}-\frac{1}{9}}{3}\)

Determine whether the function is even, odd, or neither. (a) \(f(x)=\frac{x^{2}-1}{x^{3}}\) (b) \(g(x)=\frac{x^{2}-1}{x^{4}+1}\)

Determine whether the function is even, odd, or neither. (a) \(f(x)=|x|+3\) (b) \(g(x)=-2|x|\)

Determine whether the function is even, odd, or neither. (a) \(f(x)=\frac{1}{x^{2}}\) (b) \(g(x)=\frac{2}{x^{3}}\)

Determine whether the function is even, odd, or neither. (a) \(f(x)=x+\frac{1}{x}\) (b) \(g(x)=1+\frac{1}{x}\)

Let \(f(x)=\frac{1}{x}\). Solve the following. (a) \(f\left(x^{2}\right)=1\) (b) \(-f(x)=f(x-1)\) (c) \(2 f(x-2)=f(x+3)\)

A function can be neither even nor odd. For example, consider \(f(x)=x^{3}+x^{2} .\) Can a function be both even and odd? If your answer is yes, give an example. Can you give two examples?