Chapter 3

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2}$$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=x^{3}-3 x^{2}+4$$

Fencing a Horse Corral Carol has \(2400 \mathrm{ft}\) of fencing to fence in a rectangular horse corral. (a) Find a function that models the area of the corral in terms of the width \(x\) of the corral. (b) Find the dimensions of the rectangle that maximize the area of the corral.

(a) On the same coordinate axes, sketch graphs (as accurately as possible of the functions $$y=x^{3}-2 x^{2}-x+2 \quad \text { and } \quad y=-x^{2}+5 x+2$$ (b) On the basis of your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points.

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$\overline{\bar{z}}=z$$

Graph the rational function \(f\), and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2}$$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=2 x^{3}-3 x^{2}-8 x+12$$

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z+\bar{z}\) is a real number.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{2 x^{2}-5 x}{2 x+3}$$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=x^{4}-2 x^{3}+x^{2}-9 x+2$$

Stadium Revenue \(\quad\) A baseball team plays in a stadium that holds \(55,000\) spectators. With the ticket price at \(\$ 10\), the average attendance at recent games has been \(27,000 .\) A market survey indicates that for every dollar the ticket price is lowered, attendance increases by 3000 (a) Find a function that models the revenue in terms of ticket price. (b) Find the price that maximizes revenue from ticket sales. (c) What ticket price is so high that no revenue is generated?

Recall that a function \(f\) is odd if \(f(-x)=-f(x)\) or even if \(f(-x)=f(x)\) for all real \(x\). (a) Show that a polynomial \(P(x)\) that contains only odd powers of \(x\) is an odd function. (b) Show that a polynomial \(P(x)\) that contains only even powers of \(x\) is an even function. (c) Show that if a polynomial \(P(x)\) contains both odd and even powers of \(x,\) then it is neither an odd nor an even function. (d) Express the function $$P(x)=x^{5}+6 x^{3}-x^{2}-2 x+5$$ as the sum of an odd function and an even function.

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z-\bar{z}\) is a pure imaginary number.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}-3 x^{3}+x^{2}-3 x+3}{x^{2}-3 x}$$

Find integers that are upper and lower bounds for the real zeros of the polynomial. $$P(x)=x^{5}-x^{4}+1$$

Maximizing Profit A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost \(\$ 6\) and the society sells an average of 20 per week at a price of \(\$ 10\) each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?V

(a) Graph the function \(P(x)=(x-1)(x-3)(x-4)\) and find all local extrema, correct to the nearest tenth. (b) Graph the function $$Q(x)=(x-1)(x-3)(x-4)+5$$ and use your answers to part (a) to find all local extrema, correct to the nearest tenth.

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z \cdot \bar{z}\) is a real number.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{5}}{x^{3}-1}$$

79-84 a Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=2 x^{4}+3 x^{3}-4 x^{2}-3 x+2$$

(a) Graph the function \(P(x)=(x-2)(x-4)(x-5)\) and determine how many local extrema it has. (b) If \(a

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z=\bar{z}\) if and only if \(z\) is real.

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=2 x^{4}+15 x^{3}+31 x^{2}+20 x+4$$

Maximum of a Fourth-Degree Polynomial Find the maximum value of the function $$f(x)=3+4 x^{2}-x^{4}$$ I Hint: Let \(t=x^{2} .1\)

(a) How many \(x\) -intercepts and how many local extrema does the polynomial \(P(x)=x^{3}-4 x\) have? (b) How many \(x\) -intercepts and how many local extrema does the polynomial \(Q(x)=x^{3}+4 x\) have? (c) If \(a>0,\) how many \(x\) -intercepts and how many local extrema does each of the polynomials \(P(x)=x^{3}-a x\) and \(Q(x)=x^{3}+a x\) have? Explain your answer.

Suppose that the equation \(a x^{2}+b x+c=0\) has real coefficients and complex roots. Why must the roots be complex conjugates of each other? (Think about how you would find the roots using the Quadratic Formula.)

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{x^{4}-3 x^{3}+6}{x-3}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=4 x^{4}-21 x^{2}+5$$

A market analyst working for a small appliance manufacturer finds that if the firm produces and sells \(x\) blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$ Graph the function \(P\) in an appropriate viewing rectangle and use the graph to answer the following questions. (a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, \(P(10)=-263.3\) so the firm loses \(\$ 263.30\) if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even? (b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Calculate the first 12 powers of \(i\), that is, \(i, i^{2}, i^{3}, \ldots, i^{12} .\) Do you notice a pattern? Explain how you would calculate any whole number power of \(i,\) using the pattern that you have discovered. Use this procedure to calculate \(i^{4446}\)

Graph the rational function, and find all vertical asymptotes, \(x\) - and \(y\) -intercepts, and local extrema, comect to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

The rabbit population on a small island is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where \(t\) is the time (in months) since observations of the island began. (a) When is the maximum population attained, and what is that maximum population? (b) When does the rabbit population disappear from the island?

Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$ p(t)=\frac{3000 t}{t+1} $$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population? (IMAGES CANNOT COPY)

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24$$

After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$ c(t)=\frac{30 t}{t^{2}+2} $$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6$$

A drug is administered to a patient, and the concentration of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in hours since giving the drug), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$ c(t)=\frac{5 t}{t^{2}+1} $$ Graph the function \(c\) with a graphing device. (a) What is the highest concentration of drug that is reached in the patient's bloodstream? (b) What happens to the drug concentration after a long period of time? (c) How long does it take for the concentration to drop below \(0.3 \mathrm{mg} / \mathrm{L} ?\)

Show that the polynomial does not have any rational zeros. $$P(x)=x^{3}-x-2$$

Graph the functions \(y=x^{2}\) \(y=x^{3}, y=x^{4},\) and \(y=x^{5},\) for \(-1 \leq x \leq 1,\) on the same coordinate axes. What do you think the graph of \(y=x^{100}\) would look like on this same interval? What about \(y=x^{101} ?\) Make a table of values to confirm your answers.

Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ h(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Show that the polynomial does not have any rational zeros. $$P(x)=3 x^{3}-x^{2}-6 x+12$$

As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the observer than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(s_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{n}=440 \mathrm{Hz}\). Graph the function \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically? (IMAGES CANNOT COPY)

Show that the polynomial does not have any rational zeros. $$P(x)=3 x^{3}-x^{2}-6 x+12$$

Is it possible for a third-degree polynomial to have exactly one local extremum? Can a fourth-degree polynomial have exactly two local extrema? How many local extrema can polynomials of third, fourth, fifth, and sixth-degree have? (Think about the end behavior of such polynomials.) Now give an example of a polynomial that has six local extrema.

For a camera with a lens of fixed focal length \(F\) to focus on an object located a distance \(x\) from the lens, the film must be placed a distance \(y\) behind the lens, where \(F, x,\) and \(y\) are related by $$ \frac{1}{x}+\frac{1}{y}=\frac{1}{F} $$ (See the figure.) Suppose the camera has a 55 -mm lens \((F=55)\) (a) Express \(y\) as a function of \(x\) and graph the function. (b) What happens to the focusing distance \(y\) as the object moves far away from the lens? (c) What happens to the focusing distance \(y\) as the object moves close to the lens? (IMAGES CANNOT COPY)

Show that the polynomial does not have any rational zeros. $$P(x)=x^{50}-5 x^{25}+x^{2}-1$$

Is it possible for a polynomial to have two local maxima and no local minimum? Explain.

Give an example of a rational function that has vertical asymptote \(x=3 .\) Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept 4.