Chapter 3

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=x^{3}-3 x^{2}\)

The cost \(C\) of producing \(x\) units of a product is given by \(C=0.2 x^{2}+10 x+5,\) and the average cost per unit is given by $$\bar{C}=\frac{C}{x}=\frac{0.2 x^{2}+10 x+5}{x}, \quad x>0$$ Sketch the graph of the average cost function, and estimate the number of units that should be produced to minimize the average cost per unit.

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=3 x^{3}-24 x^{2}\)

Find all real zeros of the polynomial function. $$f(x)=5 x^{4}+9 x^{3}-19 x^{2}-3 x$$

The concentration \(C\) of a chemical in the bloodstream \(t\) hours after injection into muscle tissue is given by $$C=\frac{3 t^{2}+t}{t^{3}+50}, \quad t \geq 0$$ (a) Determine the horizontal asymptote of the function and interpret its meaning in the context of the problem. (b) Use a graphing utility to graph the function and approximate the time when the bloodstream concentration is greatest. (c) Use the graphing utility to determine when the concentration is less than 0.345

The annual profit \(P\) (in dollars) of a company is modeled by a function of the form \(P=a t^{2}+b t+c,\) where \(t\) represents the year. Discuss which of the following models the company might prefer. (a) \(a\) is positive and \(t \geq-b /(2 a)\) (b) \(a\) is positive and \(t \leq-b /(2 a)\) (c) \(a\) is negative and \(t \geq-b /(2 a)\) (d) \(a\) is negative and \(t \leq-b /(2 a)\)

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=3 x^{3}-24 x^{2}\)

Find all real zeros of the polynomial function. $$g(x)=4 x^{4}-11 x^{3}-22 x^{2}+8 x$$

A driver averaged 50 miles per hour on the round trip between Baltimore, Maryland, and Philadelphia, Pennsylvania, 100 miles away. The average speeds for going and returning were \(x\) and \(y\) miles per hour, respectively. (a) Show that \(y=\frac{25 x}{x-25}\) (b) Determine the vertical and horizontal asymptotes of the function. (c) Use a graphing utility to complete the table. What do you observe? (d) Use the graphing utility to graph the function. (e) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.

Determine algebraically any point(s) of intersection of the graphs of the equations. Verify your results using the intersect feature of a graphing utility. \(\begin{aligned} x+y &=8 \\\\-\frac{2}{3} x+y &=6 \end{aligned}\)

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=-x^{6}+7 x^{3}+8\)

Find all real zeros of the polynomial function. $$f(z)=z^{4}-z^{3}-2 z-4$$

Determine algebraically any point(s) of intersection of the graphs of the equations. Verify your results using the intersect feature of a graphing utility. $$\begin{aligned} &y=3 x-10\\\ &y=\frac{1}{4} x+1 \end{aligned}$$

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(f(x)=x^{3}+3 x^{2}-9 x-27\)

Find all real zeros of the polynomial function. $$f(x)=4 x^{3}+7 x^{2}-11 x-18$$

Determine algebraically any point(s) of intersection of the graphs of the equations. Verify your results using the intersect feature of a graphing utility. $$\begin{aligned} &y=9-x^{2}\\\ &y=x+3 \end{aligned}$$

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(h(x)=x^{5}-4 x^{3}+8 x^{2}-32\)

Find all real zeros of the polynomial function. $$g(y)=2 y^{4}+7 y^{3}-26 y^{2}+23 y-6$$

Determine algebraically any point(s) of intersection of the graphs of the equations. Verify your results using the intersect feature of a graphing utility. \(y=x^{3}+2 x-1\) \(y=-2 x+15\)

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(g(t)=-\frac{1}{4} t^{4}+2 t^{2}-4\)

Find all real zeros of the polynomial function. $$h(x)=x^{5}-x^{4}-3 x^{3}+5 x^{2}-2 x$$

Determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

Sketch the graph of the function by (a) applying the Leading Coefficient Test, (b) finding the zeros of the polynomial, (c) plotting sufficient solution points, and (d) drawing a continuous curve through the points. \(g(x)=\frac{1}{10}\left(x^{4}-4 x^{3}-2 x^{2}+12 x+9\right)\)

Find all real zeros of the polynomial function. $$f(x)=4 x^{4}-55 x^{2}-45 x+36$$

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function might indicate that there should be one. $$h(x)=\frac{6-2 x}{3-x}$$

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(f(x)=x^{3}-3 x^{2}+3\)

Find all real zeros of the polynomial function. $$z(x)=6 x^{4}+33 x^{3}-69 x+30$$

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function might indicate that there should be one. $$g(x)=\frac{x^{2}+x-2}{x-1}$$

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(f(x)=-2 x^{3}-6 x^{2}+3\)

Find all real zeros of the polynomial function. $$g(x)=8 x^{4}+28 x^{3}+9 x^{2}-9 x$$

Write a set of guidelines for finding all the asymptotes of a rational function given that the degree of the numerator is not more than 1 greater than the degree of the denominator.

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(g(x)=3 x^{4}+4 x^{3}-3\)

Find all real zeros of the polynomial function. $$h(x)=x^{5}+5 x^{4}-5 x^{3}-15 x^{2}-6 x$$

Write a rational function that has the specificd characteristics. (There are many correct answers.) (a) Vertical asymptote: \(x=-2\) Slant asymptote: \(y=x+1\) Zero of the function: \(x=2\) (b) Vertical asymptote: \(x=-4\) Slant asymptote: \(y=x-2\) Zero of the function: \(x=3\)

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(h(x)=x^{4}-10 x^{2}+2\)

Find all real zeros of the polynomial function. $$f(x)=8 x^{5}+6 x^{4}-37 x^{3}-36 x^{2}+29 x+30$$

Simplify the expression. $$\left(\frac{x}{8}\right)^{-3}$$

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(f(x)=x^{4}-3 x^{3}-4 x-3\)

Find all real zeros of the polynomial function. $$g(x)=4 x^{5}+8 x^{4}-15 x^{3}-23 x^{2}+11 x+15$$

Simplify the expression. $$\left(4 x^{2}\right)^{-2}$$

(a) use the Intermediate Value Theorem and a graphing utility to find graphically any intervals of length 1 in which the polynomial function is guaranteed to have a zero, and (b) use the zero or root feature of the graphing utility to approximate the real zeros of the function. Verify your answers in part (a) by using the table feature of the graphing utility. \(f(x)=x^{3}-4 x^{2}-2 x+10\)

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$h(t)=t^{3}-2 t^{2}-7 t+2$$

Simplify the expression. $$\frac{3^{7 / 6}}{3^{1 / 6}}$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(f(x)=x^{2}(x+6)\)

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$f(s)=s^{3}-12 s^{2}+40 s-24$$

Simplify the expression. $$\frac{\left(x^{-2}\right)\left(x^{1 / 2}\right)}{\left(x^{-1}\right)\left(x^{5 / 2}\right)}$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(h(x)=x^{3}(x-3)^{2}\)

Use the zero or root feature of a graphing utility to approximate (accurate to the nearest thousandth) the zeros of the function, (b) determine one of the exact zeros and use synthetic division to verify your result, and (c) factor the polynomial completely. $$h(x)=x^{5}-7 x^{4}+10 x^{3}+14 x^{2}-24 x$$

Use a graphing utility to graph the function and find its domain and range. $$f(x)=\sqrt{6+x^{2}}$$

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\)