Chapter 2

A spotlight is to be placed on the side of a 28 -foot tall building to illuminate a bench that is 32 feet from the base of the building. The intensity \(I\) of the light at the bench is known to be \(x / d^{3},\) where \(x\) is the height of the spotlight above the ground and \(d\) is the distance from the bench to the spotlight. If the intensity is to be \(.00035,\) how high should the spotlight be?

Find an exact solution of the equation in the given open interval. For example, if the graphical approximation of a solution begins. \(3333,\) check to see whether \(1 / 3\) is the exact solution. Similarly, \(\sqrt{2} \approx 1.414 ;\) so if your approximation begins \(1.414,\) check to see whether \(\sqrt{2}\) is a solution.) $$4 x^{3}-3 x^{2}-3 x-7=0 ; \quad(1,2)$$

In 2005 Dan Wheldon won the Indianapolis 500 (mile) race. His speed was 83 mph faster than the speed of Ray Harroun, who won the race in \(1911 .\) Wheldon took 3.53 hours less than Harroun to complete the race. What was Wheldon's average speed?

In Exercises \(37-42,\) obtain a complete graph of the equation by trying various viewing windows. List a viewing window that produces this complete graph. (Many correct answers are pos. sible; consider your answer to be correct if your window shows all the features in the window given in the answer section.) $$y=7 x^{3}+35 x+10$$

A homemade loaf of bread turns out to be a perfect cube. Five slices of bread, each .6 inch thick, are cut from one end of the loaf. The remainder of the loaf now has a volume of 235 cubic inches. What were the dimensions of the original loaf?

In Exercises \(37-42,\) obtain a complete graph of the equation by trying various viewing windows. List a viewing window that produces this complete graph. (Many correct answers are pos. sible; consider your answer to be correct if your window shows all the features in the window given in the answer section.) $$y=x^{3}-5 x^{2}+5 x-6$$

Find an exact solution of the equation in the given open interval. For example, if the graphical approximation of a solution begins. \(3333,\) check to see whether \(1 / 3\) is the exact solution. Similarly, \(\sqrt{2} \approx 1.414 ;\) so if your approximation begins \(1.414,\) check to see whether \(\sqrt{2}\) is a solution.) $$8 x^{5}+7 x^{4}-x^{3}+16 x-2=0 ; \quad(0,1)$$

A homemade loaf of bread turns out to be a perfect cube. Five slices of bread, each .6 inch thick, are cut from one end of the loaf. The remainder of the loaf now has a volume of 235 cubic inches. What were the dimensions of the original loaf?

In Exercises \(37-42,\) obtain a complete graph of the equation by trying various viewing windows. List a viewing window that produces this complete graph. (Many correct answers are pos. sible; consider your answer to be correct if your window shows all the features in the window given in the answer section.) $$y=\sqrt{x^{2}}-x$$

Find an exact solution of the equation in the given open interval. For example, if the graphical approximation of a solution begins. \(3333,\) check to see whether \(1 / 3\) is the exact solution. Similarly, \(\sqrt{2} \approx 1.414 ;\) so if your approximation begins \(1.414,\) check to see whether \(\sqrt{2}\) is a solution.) $$4 x^{4}-13 x^{2}+3=0 ; \quad(1,2)$$

Find an exact solution of the equation in the given open interval. For example, if the graphical approximation of a solution begins. \(3333,\) check to see whether \(1 / 3\) is the exact solution. Similarly, \(\sqrt{2} \approx 1.414 ;\) so if your approximation begins \(1.414,\) check to see whether \(\sqrt{2}\) is a solution.) $$x^{3}+x^{2}-2 x-2=0 ; \quad(1,2)$$

In Exercises \(37-42,\) obtain a complete graph of the equation by trying various viewing windows. List a viewing window that produces this complete graph. (Many correct answers are pos. sible; consider your answer to be correct if your window shows all the features in the window given in the answer section.) $$y=-1 x^{4}+x^{3}+x^{2}+x+50$$

Deal with exponential, logarithmic, and trigonometric equations, which will be dealt with in later chapters. If you are familiar with these concepts, solve each equation graphically or numerically. $$10^{x}-\frac{1}{4} x=28$$

Deal with exponential, logarithmic, and trigonometric equations, which will be dealt with in later chapters. If you are familiar with these concepts, solve each equation graphically or numerically. $$e^{x}-6 x=5$$

In Exercises \(43-46,\) use technology to construct a scatter plot and a line graph of the data. Last year's electric bills for one of the authors are shown in the table. Let \(x=3\) correspond to March, \(x=4\) to April, etc. $$\begin{array}{|l|c|} \hline \text { Month } & \text { Bill ( } \$ \text { ) } \\ \hline \text { March } & 61 \\ \hline \text { April } & 50 \\ \hline \text { May } & 116 \\ \hline \text { June } & 187 \\ \hline \text { July } & 149 \\ \hline \text { August } & 182 \\ \hline \text { September } & 77 \\ \hline \end{array}$$

Deal with exponential, logarithmic, and trigonometric equations, which will be dealt with in later chapters. If you are familiar with these concepts, solve each equation graphically or numerically. $$x+\sin \left(\frac{x}{2}\right)=4$$

In Exercises \(43-46,\) use technology to construct a scatter plot and a line graph of the data. The table shows the population of Kansas City, Missouri in various years." Let \(x=0\) correspond to 1950 . $$\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & 1950 & 1960 & 1970 & 1980 & 1990 & 2004 \\ \hline \text { Population } & 457 & 476 & 507 & 448 & 435 & 444 \\ \hline \end{array}$$

Deal with exponential, logarithmic, and trigonometric equations, which will be dealt with in later chapters. If you are familiar with these concepts, solve each equation graphically or numerically. $$x^{3}+\cos \left(\frac{x}{3}\right)=5$$

In Exercises \(43-46,\) use technology to construct a scatter plot and a line graph of the data. The average monthly rainfall (in inches) in Cleveland, Ohio (based on a thirty year average) is shown in the table." Let \(x=1\) correspond to January, \(x=3\) to March, etc. $$\begin{array}{|l|c|} \hline \text { Month } & \text { Rainfall } \\ \hline \text { January } & 2.5 \\ \hline \text { March } & 2.9 \\ \hline \text { May } & 3.5 \\ \hline \text { July } & 3.5 \\ \hline \text { September } & 3.8 \\ \hline \text { November } & 3.4 \\ \hline \end{array}$$

Deal with exponential, logarithmic, and trigonometric equations, which will be dealt with in later chapters. If you are familiar with these concepts, solve each equation graphically or numerically. $$\ln x-x^{2}+3=0$$

(a) Graph \(y=-3(x-3)^{4}+8\) in the standard window. (b) To see exactly which points the calculator actually graphed, change the graphing mode to Dot (or Draw Dot or Plot ) in the menu/submenu list below, and graph again. TI- \(84+:\) MODE TI-86: GRAPH/FORMAT TI- \(89: Y=/\) STYLE Casio: SETUP/DRAW TYPE HP-39gs: uncheck "connect" on the second page of the PLOT SETUP menu. (c) Why does the graph in part (b) look "solid" at the top, but consists of isolated points elsewhere?

According to data from the U.S. Department of Education, the average cost \(y\) of tuition and fees at four-year public colleges and universities in year \(x\) is approximated by $$ y=\sqrt{180,115 x^{2}+2,863,851 x+11,383,876} $$ where \(x=0\) corresponds to \(2000 .\) If this model continues to be accurate, in what year will tuition and fees reach \(\$ 7000 ?\) Round your answer to the nearest year.

(a) Graph \(y=3 x^{3}-2 x^{2}+6\) in the standard window. (b) Use trace to move to a point whose \(x\) -coordinate is close to 1 (c) Set the zoom factors of your calculator to \(10 .\) Zoom-in once or twice. Does the graph appear to be a straight line near the point? (d) Repeat parts (a) \(-(c)\) at lowest point to the right of the \(y\) -axis. Is the result the same? If not, keep zooming in until it is (at each stage move the flashing cursor up or down, so it is on the graph). (e) What do parts (a) - (d) suggest about the graph?

According to data from the U.S. Department of Health and Human Services, the cumulative number \(y\) of AIDS cases (in thousands) as of year \(x\) is approximated by \(y=.004 x^{3}-1.367 x^{2}+54.35 x+569.72 \quad(0 \leq x<11)\) where \(x=0\) corresponds to 1995 (a) When did the cumulative number of cases reach \(944,000 ?\) (b) If this model remains accurate after \(2006,\) in what year will the cumulative number of cases reach 1.1 million?

The enrollment in public high schools (in millions of students ) in year \(x\) is approximated by $$\begin{aligned} y=&-.000035606 x^{4}+.0021 x^{3}-.02714 x^{2} &-.12059 x & \\ &+14.2996 &(0 \leq x<35) \end{aligned}$$ where \(x=0\) corresponds to \(1975 .^{*}\) During the current century, when was enrollment 13.9 million students?

In Exercises \(49-54,\) use your algebraic knowledge to state whether or not the two equations have the same graph. Confirm your answer by graphing the equations in the standard window. $$y=\sqrt{x^{2}} \text { and } y=|x|$$

In Exercises \(49-54,\) use your algebraic knowledge to state whether or not the two equations have the same graph. Confirm your answer by graphing the equations in the standard window. $$y=\sqrt{x^{2}+6 x+9} \text { and } y=|x+3|$$

According to the U.S. Centers for Medicare and Medicaid Services, total medical expenditures (in billions of dollars) in the United States in year \(x\) are expected to be given by $$y=-.035 x^{4}+1.01 x^{3}-4.91 x^{2}+126.94 x+1309.6$$ where \(x=0\) corresponds to \(2000 .\) When will expenditures be \(\$ 2.6\) trillion?

In Exercises \(49-54,\) use your algebraic knowledge to state whether or not the two equations have the same graph. Confirm your answer by graphing the equations in the standard window. $$y=\frac{1}{x^{2}+2} \text { and } y=\frac{1}{x^{2}}+\frac{1}{2}$$

(a) How many real solutions does the equation $$ .2 x^{5}-2 x^{3}+1.8 x+k=0 $$ have when \(k=0 ?\) (b) How many real solutions does it have when \(k=1 ?\) (c) Is there a value of \(k\) for which the equation has just one real solution? (d) Is there a value of \(k\) for which the equation has no real solutions?

(a) Confirm the accuracy of the factorization \(x^{2}-5 x+6=(x-2)(x-3)\) graphically. [Hint: Graph \(y=x^{2}-5 x+6\) and \(y=(x-2)(x-3)\) on the same screen. If the factorization is correct, the graphs will be identical (which means that you will see only a single graph on the screen).l (b) Show graphically that \((x+5)^{2} \neq x^{2}+5^{2} .\) [ Hint: Graph \(y=(x+5)^{2}\) and \(y=x^{2}+5^{2}\) on the same screen. If the graphs are different, then the two expressions cannot be equal. \(]\)

True or False. In Exercises \(56-58,\) use the technique of Exercise 55 to determine graphically whether the given state. ment is possibly true or definitely false. ( We say "possibly true " because nwo graphs that appear identical on a calculator screen may actually differ by small amounts or at places not shown in the window.) $$(1-x)^{6}=1-6 x+15 x^{2}-20 x^{3}+15 x^{4}-6 x^{5}+x^{6}$$

A toy rocket is shot straight up from ground level and then falls back to earth; wind resistance is negligible. Use your calculator to determine which of the following equations has a graph whose portion above the \(x\) -axis provides the most plausible model of the path of the rocket. (a) \(y=1(x-3)^{3}-1 x^{2}+5\) (b) \(y=-x^{4}+16 x^{3}-88 x^{2}+192 x\) (c) \(y=-16 x^{2}+117 x\) (d) \(y=16 x^{2}-3.2 x+16\) (e) \(y=-(.1 x-3)^{6}+600\)

Monthly profits at DayGlo Tee Shirt Company appear to be given by the equation $$ y=-.00027(x-15,000)^{2}+60,000 $$ where \(x\) is the number of shirts sold that month and \(y\) is the profit. DayGlo's maximum production capacity is 15,000 shirts per month. (a) If you plan to graph the profit equation, what range of \(x\) values should you use? [Hint: You can't make a negative number of shirts.] (b) The president of DayGlo wants to motivate the sales force (who are all in the profit-sharing plan), so he asks you to prepare a graph that shows DayGlo's profits increasing dramatically as sales increase. Using the profit equation and the \(x\) range from part (a), what viewing window would be suitable? (c) The City Council is talking about imposing more taxes. The president asks you to prepare a graph showing that DayGlo's profits are essentially flat. Using the profit equation and the \(x\) range from part (a), what viewing window would be suitable?

In each of the applied situations in Exercises \(61-64\), find an appropriate viewing window for the equation (that is, a window that includes all the points relevant to the problem but does not include large regions that are not relevant to the prob. lem, and has easily readable tick marks on the axes). Explain why you chose this window. See the Hint in Exercise \(60(a)\) Beginning in 1905 the deer population in a region of Arizona rapidly increased because of a lack of natural predators. Eventually food resources were depleted to such a degree that the deer population completely died out. In the equation \(y=-125 x^{5}+3.125 x^{4}+4000, y\) is the number of deer in year \(x,\) where \(x=0\) corresponds to 1905

Beginning in 1905 the deer population in a region of Arizona rapidly increased because of a lack of natural predators. Eventually food resources were depleted to such a degree that the deer population completely died out. In the equation \(y=-125 x^{5}+3.125 x^{4}+4000, y\) is the number of deer in year \(x,\) where \(x=0\) corresponds to 1905 A cardiac test measures the concentration \(y\) of a dye \(x\) seconds after a known amount is injected into a vein near the heart. In a normal heart $$ y=-.006 x^{4}+.14 x^{3}-.053 x^{2}+179 x $$

In each of the applied situations in Exercises \(61-64\), find an appropriate viewing window for the equation (that is, a window that includes all the points relevant to the problem but does not include large regions that are not relevant to the prob. lem, and has easily readable tick marks on the axes). Explain why you chose this window. See the Hint in Exercise \(60(a)\) The concentration of a certain medication in the blood. stream at time \(x\) hours is approximated by the equation $$ y=\frac{375 x}{.1 x^{3}+50} $$

The total resources (in billions of dollars) of the Pension Benefit Guaranty Corporation, the government agency that insures pensions, is approximated by $$ y=-.279 x^{2}+4.006 x+28.412 \quad(4 \leq x \leq 20) $$ where \(x=4\) corresponds to 2004 (a) When are resources the greatest? (b) Use the trace feature to find the approximate time when the Corporation will run out of money.

In Exercises \(69-72,\) graph all four equations on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship of graphs (b), (c), and (d) to graph (a)? (a) \(y=x^{2}\) (b) \(y=x^{2}+5\) (c) \(y=x^{2}-5\) (d) \(y=x^{2}-2\)

In Exercises \(69-72,\) graph all four equations on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship of graphs (b), (c), and (d) to graph (a)? (a) \(y=\sqrt{x}\) (b) \(y=\sqrt{x-3}\) (c) \(y=\sqrt{x+3}\) (d) \(y=\sqrt{x-6}\)

In Exercises \(69-72,\) graph all four equations on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship of graphs (b), (c), and (d) to graph (a)? (a) \(y=\sqrt{x}\) (b) \(y=2 \sqrt{x}\) (c) \(y=3 \sqrt{x}\) (d) \(y=\frac{1}{2} \sqrt{x}\)

In Exercises \(69-72,\) graph all four equations on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship of graphs (b), (c), and (d) to graph (a)? (a) \(y=x^{2}\) (b) \(y=-x^{2}\) (c) \(y=-\frac{1}{2} x^{2}\) (d) \(y=-2 x^{2}\)

In Exercises \(73-75,\) graph the two given equations and the equation \(y=x\) on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship between graphs (a) and (b)? (a) \(y=x^{3}\) (b) \(y=\sqrt[3]{x}\)

In Exercises \(73-75,\) graph the two given equations and the equation \(y=x\) on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship between graphs (a) and (b)? (a) \(y=\frac{1}{2} x^{3}-4\) (b) \(y=\sqrt[3]{2 x+8}\)

In Exercises \(73-75,\) graph the two given equations and the equation \(y=x\) on the same screen, using a sufficiently large square viewing window, and answer this question: What is the geometric relationship between graphs (a) and (b)? (a) \(y=\frac{1}{2} x^{3}-4\) (b) \(y=\sqrt[3]{2 x+8}\)

Put your calculator in radian mode, and use the viewing window given by \(0 \leq x \leq 6.28\) and \(-2 \leq y \leq 2 .\) (a) Graph \(y=\sin x\) (b) Graph \(y=\sin (2 x)\) (c) Graph \(y=\sin (3 x)\) (d) On the basis of parts (a)-(c), what do you think the graphs of \(y=\sin (4 x), y=\sin (5 x), y=\sin (6 x),\) and so on, will look like? Use the calculator to verify your answer. (e) On the basis of part (d), what do you think the graphs of \(y=\sin (50 x)\) and \(y=\sin (100 x)\) will look like? What does a calculator display instead? What might explain the graphs of the calculator?