Chapter 3

An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200 -meter single-lane running track. (a) Draw a diagram that illustrates the problem. Let \(x\) and \(y\) represent the length and width of the rectangular region, respectively. (b) Determine the radius of the semicircular ends of the track. Determine the distance, in terms of \(y\), around the inside edge of the two semicircular parts of the track. (c) Use the result of part (b) to write an equation, in terms of \(x\) and \(y,\) for the distance traveled in one lap around the track. Solve for \(y\) (d) Use the result of part (c) to write the area \(A\) of the rectangular region as a function of \(x\) (e) Use a graphing utility to graph the area function from part (d). Use the graph to approximate the dimensions that will produce a rectangle of maximum area.

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=-2 x^{4}+5 x^{2}-x-1\)

Divide using long division. $$\left(4 x^{5}+3 x^{3}-10\right) \div(2 x+3)$$

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=3 x^{4}+5 x^{3}-6 x^{2}+8 x-3$$

Use a graphing utility to graph the rational function. Determine the domain of the function and identify any asymptotes. $$y=\frac{12-2 x-x^{2}}{2(4+x)}$$

Use the given zero to find all the zeros of the function. Function $$f(x)=25 x^{3}-55 x^{2}-54 x-18$$ Zero $$\frac{1}{5}(-2+\sqrt{2} i)$$

Use a graphing utility to graph the function and approximate (accurate to three decimal places) any real zeros and relative extrema. \(f(x)=3 x^{5}-2 x^{2}-x+1\)

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$g(x)=4 x^{3}-5 x+8$$

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{x^{2}-5 x+4}{x^{2}-4}$$

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22$$

The height \(y\) (in feet) of a punted football is approximated by \(y=-\frac{16}{2025} x^{2}+\frac{9}{5} x+\frac{3}{2}\) where \(x\) is the horizontal distance (in feet) from where the football is punted. (See figure.) (a) Use a graphing utility to graph the path of the football. (b) How high is the football when it is punted? (Hint: Find \(y\) when \(x=0 .\) ) (c) What is the maximum height of the football? (d) How far from the punter does the football strike the ground?

Find a polynomial function that has the given zeros. (There are many correct answers.) \(0,7\)

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{x^{2}-2 x-8}{x^{2}-9}$$

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$f(x)=x^{3}+4 x^{2}+14 x+20$$

The path of a diver is approximated by \(y=-\frac{4}{9} x^{2}+\frac{24}{9} x+12\) where \(y\) is the height (in feet) and \(x\) is the horizontal distance (in feet) from the end of the diving board (see figure). What is the maximum height of the diver?

Find a polynomial function that has the given zeros. (There are many correct answers.) \(-2,5\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=x^{3}+x^{2}-4 x-4$$

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{2 x^{2}-5 x+2}{2 x^{2}-x-6}$$

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$h(x)=8 x^{3}-14 x^{2}+18 x-9$$

Find a polynomial function that has the given zeros. (There are many correct answers.) \(0,-2,-4\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=-3 x^{3}+20 x^{2}-36 x+16$$

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{3 x^{2}-8 x+4}{2 x^{2}-3 x-2}$$

(a) use a graphing utility to find the real zeros of the function, and then (b) use the real zeros to find the exact values of the imaginary zeros. $$f(x)=25 x^{3}-55 x^{2}-54 x-18$$

The monthly revenue \(R\) (in thousands of dollars) from the sales of a digital picture frame is approximated by \(R(p)=-10 p^{2}+1580 p,\) where \(p\) is the price per unit (in dollars). (a) Find the monthly revenues for unit prices of \(\$ 50\) \(\$ 70,\) and \(\$ 90\) (b) Find the unit price that will yield a maximum monthly revenue. (c) What is the maximum monthly revenue?

Find a polynomial function that has the given zeros. (There are many correct answers.) \(0,1,6\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=-2 x^{4}+13 x^{3}-21 x^{2}+2 x+8$$

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{2 x^{3}-x^{2}-2 x+1}{x^{2}+3 x+2}$$

A football is kicked off the ground with an initial upward velocity of 48 feet per second. The football's height \(h\) (in feet) is given by \(h(t)=-16 t^{2}+48 t, \quad 0 \leq t \leq 3\) where \(t\) is the time (in seconds). Does the football reach a height of 50 feet? Explain.

For selected years from 1955 through \(2010,\) the annual per capita consumption \(C\) of cigarettes by Americans (ages 18 and older) can be modeled by \(C(t)=-1.39 t^{2}+36.5 t+3871, \quad 5 \leq t \leq 60\) where \(t\) is the year, with \(t=5\) corresponding to 1955 (a) Use a graphing utility to graph the model. (b) Use the graph of the model to approximate the year when the maximum annual consumption of cigarettes occurred. Approximate the maximum average annual consumption. (c) Beginning in \(1966,\) all cigarette packages were required by law to carry a health warning. Do you think the warning had any effect? Explain. (d) In \(2010,\) the U.S. population (ages 18 and older) was \(234,564,000 .\) Of those, about 45,271,000 were smokers. What was the average annual cigarette consumption per smoker in \(2010 ?\) What was the average daily cigarette consumption per smoker?

Find a polynomial function that has the given zeros. (There are many correct answers.) \(4,-3,3,0\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=4 x^{4}-17 x^{2}+4$$

Find all vertical asymptotes, horizontal asymptotes, slant asymptotes, and holes in the graph of the function. Then use a graphing utility to verify your results. $$f(x)=\frac{2 x^{3}+x^{2}-8 x-4}{x^{2}-3 x+2}$$

The demand equation for a microwave is \(p=140-0.001 x,\) where \(p\) is the unit price (in dollars) of the microwave and \(x\) is the number of units produced and sold. The cost equation for the microwave is \(C=40 x+150,000,\) where \(C\) is the total cost (in dollars) and \(x\) is the number of units produced. The total profit \(P\) obtained by producing and selling \(x\) units is given by \(P=R-C=x p-C .\) Is there a price \(p\) that yields a profit of \(\$ 3\) million? Explain.

The population \(P\) of Germany (in thousands) from 2000 through 2013 can be modeled by \(P(t)=-14.82 t^{2}+95.9 t+82,276, \quad 0 \leq t \leq 13\) where \(t\) is the year, with \(t=0\) corresponding to 2000 (a) According to the model, in what year did Germany have its greatest population? What was the population? (b) According to the model, what will Germany's population be in the year \(2075 ?\) Is this result reasonable? Explain.

Find a polynomial function that has the given zeros. (There are many correct answers.) \(-2,-1,0,1,2\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=32 x^{3}-52 x^{2}+17 x+3$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{1}{x+5}+\frac{4}{x}$$

Decide whether the statement is true or false. Justify your answer. It is possible for a third-degree polynomial function with integer coefficients to have no real zeros.

Find a polynomial function that has the given zeros. (There are many correct answers.) \(1+\sqrt{2}, 1-\sqrt{2}\)

(a) use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of \(f\) (b) list the possible rational zeros of \(f,\) (c) use a graphing utility to graph \(f\) so that some of the possible zeros in parts (a) and (b) can be disregarded, and (d) determine all the real zeros of \(f\). $$f(x)=x^{4}-x^{3}-29 x^{2}-x-30$$

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{1}{x-2}-\frac{5}{x}$$

Decide whether the statement is true or false. Justify your answer. If \([x+(4+3 i)]\) is a factor of a polynomial function \(f\) with real coefficients, then \([x-(4+3 i)]\) is also a factor of \(f\)

Determine whether the statement is true or false. Justify your answer. The function \(f(x)=a(x-5)^{2}\) has exactly one \(x\) -intercept for any nonzero value of \(a\)

Find a polynomial function that has the given zeros. (There are many correct answers.) \(4+\sqrt{3}, 4-\sqrt{3}\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=x^{4}-4 x^{3}+15\) Upper bound: \(x=4\) Lower bound: \(x=-1\)

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{2}{x+2}-\frac{3}{x-1}$$

Determine whether the statement is true or false. Justify your answer. The functions \(f(x)=3 x^{2}+6 x+7 \quad\) and \(g(x)=3 x^{2}+6 x-1\) have the same vertex.

Find a polynomial function that has the given zeros. (There are many correct answers.) \(2,2+\sqrt{5}, 2-\sqrt{5}\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f .\) Then find all real zeros of the function. \(f(x)=2 x^{3}-3 x^{2}-12 x+8\) Upper bound: \(x=4\) Lower bound: \(x=-3\)

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=\frac{6}{x+3}-\frac{1}{x+4}$$