Chapter 4

Does there exist a continuous odd function that is always increasing and whose graph passes through the points \((-3,-4)\) and \((2,5) ?\) Bxplain.

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{4 x^{2}}{2 x-1} $$

Is there an even function whose domain is all real numbers and that is always decreasing? Explain.

Complete the following. (a) Find any slant or vertical asymptotes. (b) Graph \(y=f(x) .\) Show all asymptotes. $$ f(x)=\frac{4 x^{2}+x-2}{4 x-3} $$

A cardboard box with no top and a square base is being constructed and must have a volume of 108 cubic inches. Let \(x\) be the length of a side of its base in inches. (a) Write a formula \(A(x)\) that calculates the outside surface area in square feet of the box. (b) If cardboard costs \(\$ 0.10\) per square foot, write a formula \(C(x)\) that gives the cost in dollars of the cardboard in the box. (c) Find the dimensions of the box that would minimize the cost of the cardboard.

Sketch a graph of a continuous function with an absolute minimum of \(-3\) at \(x=-2\) and a local minimum of \(-1\) at \(x=2\)

Solve the equation. $$ \frac{4}{x+2}=-4 $$

A cost-benefit function \(C\) computes the cost in millions of dollars of implementing a city recycling project when \(x\) percent of the citizens participate, where \(C(x)=100-x\) (a) Graph \(C\) in \([0,100,10]\) by \([0,10,1]\). Interpret the graph as \(x\) approaches 100 (b) If \(75 \%\) participation is expected, determine the cost for the city. (c) The city plans to spend \(\$ 5\) million on this recycling project. Estimate the percentage of participation that can be expected.

Sketch a graph of a continuous function with no absolute extrema but with a local minimum of \(-2\) at \(x=-1\) and a local maximum of 2 at \(x=1\)

Solve the equation. $$ \frac{3}{2 x+1}=-1 $$

Solve the equation. $$ \frac{x+1}{x}=2 $$

Sketch a graph of a continuous function with a local maximum of 2 at \(x=-1\) and a local maximum of 0 at \(x=1\)

Solve the equation. $$ \frac{2 x}{x-3}=-4 $$

Solve the equation. $$ \frac{1-x}{3 x-1}=-\frac{3}{5} $$

If a parking garage attendant can wait on 3 vehicles per minute and vehicles are leaving the ramp at \(x\) vehicles per minute, then the average wait in minutes for a car trying to exit is given by the formula \(f(x)=\frac{1}{3-x^{2}}\) (a) Solve the three-part inequality \(5 \leq \frac{1}{3-x} \leq 10\) (b) Interpret your result from part (a).

Solve the equation. $$ \frac{3-2 x}{x+2}=12 $$

The coefficient of friction \(x\) measures the friction between the tires of a car and the road, where \(0

The monthly average high temperature in degrees Fahrenheit at Daytona Beach, Florida, can be approximated by \(f(x)=0.0145 x^{4}-0.426 x^{3}+3.53 x^{2}-6.22 x+72\) where \(x=1\) corresponds to January, \(x=2\) to February, and so on. Estimate graphically when the monthly average high temperature is \(75^{\circ} \mathrm{F}\) or more.

Graph \(f\). Use the steps for graphing a rational function described in this section. $$ f(x)=\frac{x+3}{2 x-4} $$

A cubical box is being manufactured to hold 213 cubic inches. If this measurement can vary between 212.8 cubic inches and 213.2 cubic inches inclusive, by how much can the length \(x\) of a side of the cube vary?

If the graph of \(y=f(x)\) is increasing on \([1,4],\) then where is the graph of \(y=f(x+1)-2\) increasing? Where is the graph of \(y=-f(x-2)\) decreasing?

A cylindrical aluminum can is being manufactured so that its height \(h\) is 8 centimeters more than its radius \(r\). Estimate values for the radius (to the nearest hundredth) that result in the can having a volume between 1000 and 1500 cubic centimeters inclusive.

If the graph of \(f\) is decreasing on \([0, \infty),\) then what can be said about the graph of \(y=f(-x)+17\) the graph of \(y=-f(x)-1 ?\)

Find the constant of proportionality \(k\) $$ y=\frac{k}{x}, \text { and } y=2 \text { when } x=3 $$

Graph \(f\). Use the steps for graphing a rational function described in this section. $$ f(x)=\frac{x^{2}+2 x+1}{x^{2}-x-6} $$

Find the constant of proportionality \(k\) $$ y=\frac{k}{x^{2}} \text { and } y=\frac{1}{4} \text { when } x=8 $$

Graph \(f\). Use the steps for graphing a rational function described in this section. $$ f(x)=\frac{3 x^{2}+3 x-6}{x^{2}-x-12} $$

Find the constant of proportionality \(k\) $$ y=k x^{3}, \text { and } y=64 \text { when } x=2 $$

If two parking attendants can wait on 8 vehicles per minute and vehicles are leaving the parking garage randomly at an average rate of \(x\) vehicles per minute, then the average time \(T\) in minutes spent waiting in line and paying the attendant is given by the formula \(T(x)=-\frac{1}{x-8},\) where \(0 \leq x<8 .\) A graph of \(T\) is shown in the figure. (a) Evaluate \(T(4)\) and \(T(7.5) .\) Interpret the results. (b) What happens to the wait as vehicles arrive at an average rate that approaches 8 cars per minute?

Find the constant of proportionality \(k\) $$ y=k x^{3 / 2}, \text { and } y=96 \text { when } x=16 $$

From 1900 to \(2005,\) the birth rate (births per 1000 people) \(x\) years after 1900 can be approximated by \(f(x)=-0.0000285 x^{3}+0.0057 x^{2}-0.48 x+34.4\) (Source: National Center for Health Statistics.) (a) Bvaluate \(f(65)\) and interpret your result. (b) If the domain of \(f\) is \(1900 \leq x \leq 2005,\) identify the absolute extrema. Interpret cach.

If the parking attendants can wait on 5 vehicles per minute, the average time \(T\) in mimutes spent waiting in line and paying the attendant becomes \(T(x)=-\frac{1}{x-5}\) (a) What is a reasonable domain for \(T ?\) (b) Graph \(y=T(x) .\) Include any vertical asymptotes. (c) Explain what happens to \(T(x)\) as \(x \rightarrow 5\)

Solve the variation problem. Suppose \(T\) varies directly as the \(\frac{3}{2}\) power of \(x .\) When \(x=4, T=20 .\) Find \(T\) when \(x=16\)

The U.S. consumption of energy from 1950 to 1980 can be modeled by \(f(x)=-0.00113 x^{3}+0.0408 x^{2}-0.0432 x+7.66\) where \(x=0\) corresponds to 1950 and \(x=30\) to 1980 Consumption is measured in quadrillion Btu. (Source: Department of Energy.) (a) Evaluate \(f(5)\) and interpret the result. (b) Graph \(f\) in \([0,30,5]\) by \([6,16,1]\). Describe the cnergy usage during this time period. (c) Approximate the local maximum and interpret it.

Suppose that a parking attendant can wait on 40 cars per hour and that cars arrive randomly at a rate of \(x\) cars per hour. Then the average number of cars waiting in line can be cstimated by $$ N(x)=\frac{x^{2}}{1600-40 x} $$ (a) Evaluate \(N(20)\) and \(N(39)\) (b) Explain what happens to the length of the line as \(x\) approaches 40 (c) Find any vertical asymptotes of the graph of \(N\).

Solve the variation problem. Suppose \(y\) varies directly as the second power of \(x .\) When \(x=3, y=10.8 .\) Find \(y\) when \(x=1.5\)

The U.S. consumption of natural gas from 1965 to 1980 can be modeled by $$ \begin{array}{c} f(x)=0.0001234 x^{4}-0.005689 x^{3}+0.08792 x^{2} \\ -0.5145 x+1.514 \end{array} $$ where \(x=6\) corresponds to 1966 and \(x=20\) to 1980 Consumption is measured in trillion cubic feet. (Source: Department of Energy.) (a) Evaluate \(f(10)\) and interpret the result. (b) Graph \(f\) in \([6,20,5]\) by \([0.4,0.8,0.1] .\) Describe the energy usage during this time period. (c) Determine the local extrema and interpret the results.

Suppose that a construction zone can allow 50 cars per hour to pass through and that cars arrive randomly at a rate of \(x\) cars per hour. Then the average number of cars waiting in line to get through the construction zone can be estimated by $$ N(x)=\frac{x^{2}}{2500-50 x} $$ (a) Evaluate \(N(20), N(40),\) and \(N(49)\) (b) Explain what happens to the length of the line as \(x\) approaches \(50 .\) (c) Find any vertical asymptotes of the graph of \(N .\)

Solve the variation problem. Let \(y\) be inversely proportional to \(x\). When \(x=6, y=5\). Find \(y\) when \(x=15\)

Suppose that an insect population in millions is modeled by \(f(x)=\frac{10 x+1}{x+1}\) where \(x \geq 0\) is in months. (a) Graph \(f\) in \([0,14,1]\) by \([0,14,1] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population over time? (d) Interpret the horizontal asymptote.

Solve the variation problem. Let \(z\) be inversely proportional to the third power of \(t\) When \(t=5, z=0.08 .\) Find \(z\) when \(t=2\)

When a projectile is shot into the air, it attains a maximum height and then falls back to the ground. Suppose that \(x=0\) corresponds to the time when the projectile's height is maximum. If air resistance is ignored, its height \(h\) above the ground at any time \(x\) may be modeled by \(h(x)=-16 x^{2}+h_{\max }\) where \(h_{\max }\) is the projectile's maximum height above the ground. Height is measured in feet and time in seconds. Let \(h_{\max }=400\) feet. (a) Evaluate \(h(-2)\) and \(h(2) .\) Interpret these results. (b) Evaluate \(h(-5)\) and \(h(5) .\) Interpret these results. (c) Graph \(h\) for \(-5 \leq x \leq 5 .\) Is \(h\) even or odd? (d) How do \(h(x)\) and \(h(-x)\) compare when \(-5 \leq x \leq 5 ?\) What does this result indicate?

Suppose that the population of a species of fish (in thousands) is modeled by \(f(x)=\frac{x+10}{0.5 x^{2}+1},\) where \(x \geq 0\) is in years. (a) Graph \(f\) in \([0,12,1]\) by \([0,12,1] .\) What is the horizontal asymptote? (b) Determine the initial population. (c) What happens to the population of this fish? (d) Interpret the horizontal asymptote.

Assume that the constant of proportionality is positive. Let \(y\) be inversely proportional to \(x\). If \(x\) doubles, what happens to \(y ?\)

Explain the difference between a local and an absolute maximum. Are extrema \(x\) -values or \(y\) -values?

Assume that the constant of proportionality is positive. Let \(y\) vary inversely as the second power of \(x\). If \(x\) doubles, what happens to \(y ?\)

Describe ways to determine if a polynomial function is odd, even, or neither. Give examples.

Probability\(\quad\) A container holds \(x\) balls numbered 1 through \(x .\) Only one ball has the winning number. (a) Find a function \(f\) that computes the probability, or likelihood, of not drawing the winning ball. (b) What is the domain of \(f ?\) (c) What happens to the probability of not drawing the winning ball as the number of balls increases? (d) Interpret the horizontal asymptote of the graph of \(f\)

Assume that the constant of proportionality is positive. Suppose \(y\) varies directly as the third power of \(x .\) If \(x\) triples, what happens to \(y ?\)

If an odd function \(f\) has one local maximum of 5 at \(x=3,\) then what else can be said about \(f ?\) Explain.