Chapter 5

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=\frac{2}{e^{x}+e^{-x}}$$

(a) Find the limits of the function as \(x \rightarrow+\infty\) and \(x \rightarrow-\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x e^{-2 x}$$

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=x^{2} e^{-2 x}$$

Use a graphing utility to make a conjecture about the relative extrema of \(f,\) and then check your conjecture using either the first or second derivative test. $$f(x)=10 \ln x-x$$

(a) Find the limits of the function as \(x \rightarrow+\infty\) and \(x \rightarrow-\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x^{2} e^{2 x}$$

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f .\) Check that your estimates are consistent with the graph of \(f\). f(x)=x^{4}-24 x^{2}+12 x, \quad-5 \leq x \leq 5

Use a graphing utility to generate the graphs of \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval, and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f .\) Check that your estimates are consistent with the graph of \(f\). $$f(x)=\sin \frac{1}{2} x \cos x, \quad-\pi / 2 \leq x \leq \pi / 2$$

(a) Find the limits of the function as \(x \rightarrow 0^{+}\) and \(x \rightarrow+\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x \ln x$$

(a) Find the limits of the function as \(x \rightarrow 0^{+}\) and \(x \rightarrow+\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$x^{2} \ln x$$

In each part, determine whether the statement is true or false. If it is false, find functions for which the statement fails to Hold. (a) If \(f\) and \(g\) are increasing on an interval, then so is \(f+g .\) (b) If \(f\) and \(g\) are increasing on an interval, then so is \(f \cdot g\).

(a) Find the limits of the function as \(x \rightarrow 0^{+}\) and \(x \rightarrow+\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$\frac{\ln x}{x^{2}}$$

In each part, find functions \(f\) and \(g\) that are increasing on \((-\infty,+\infty)\) and for which \(f-g\) has the stated property. (a) \(f-g\) is decreasing on \((-\infty,+\infty)\) (b) \(f-g\) is constant on \((-\infty,+\infty)\) (c) \(f-g\) is increasing on \((-\infty,+\infty)\)

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime}\) over the stated interval. Use those graphs to make a conjecture about the locations and nature of the relative extrema of \(f\) and check your conjecture by graphing \(f\). $$f(x)=\frac{x^{3}-8 x+7}{\sqrt{x^{2}+1}}$$

(a) Find the limits of the function as \(x \rightarrow 0^{+}\) and \(x \rightarrow+\infty .\) (b) Give a complete graph of the function, and identify the location of all relative extrema and inflection points. Check your work with a graphing utility. $$\frac{\ln x}{\sqrt{x}}$$

(a) Prove that a general cubic polynomial $$f(x)=a x^{3}+b x^{2}+c x+d \quad(a \neq 0)$$has exactly one inflection point. (b) Prove that if a cubic polynomial has three \(x\) -intercepts. then the inflection point occurs at the average value of the intercepts. (c) Use the result in part ( \(b\) ) to find the inflection point of the cubic polynomial \(f(x)=x^{3}-3 x^{2}+2 x,\) and check your result by using \(f^{\prime \prime}\) to determine where \(f\) is concave up and concave down.

In each part, find \(k\) so that \(f\) has a relative extremum at the point \(x=3\) (a) \(f(x)=x^{2}+\frac{k}{x}\) (b) \(f(x)=\frac{x}{x^{2}+k}\)

In each part: (i) Make a conjecture about the behavior of the graph in the vicinity of its \(x\) -intercepts. (ii) Make a rough sketch of the graph based on your conjecture and the limits of the polynomials as \(x \rightarrow+\infty\) and \(x \rightarrow-\infty .\) (iii) Compare your sketch to the graph generated with a graphing utility. (a) \(y=x(x-1)(x+1) \quad\) (b) \(y=x^{2}(x-1)^{2}(x+1)^{2}\) (d) \(y=x(x-1)^{5}(x+1)^{4}\) (c) \(y=x^{2}(x-1)^{2}(x+1)^{3}\)

From Exercise \(49,\) the polynomial \(f(x)=x^{3}+b x^{2}+1\) has one inflection point. Use a graphing utility to reach a conclusion about the effect of the constant \(b\) on the location of the inflection point. Use \(f^{\prime \prime}\) to explain what you have observed graphically.

Functions of the form $$f(x)=c x^{n} e^{-x}, \quad x>0$$ where \(n\) is a positive integer and \(c=1 / n !\), arise in the statistical study of traffic flow. (a) Use a graphing utility to generate the graph of \(f\) for \(n=2,3,4,\) and \(5,\) and make a conjecture about the number and locations of the relative extrema of \(f\) (b) Confirm your conjecture using the first derivative test.

Sketch the graph of \(y=(x-a)^{m}(x-b)^{n}\) for the stated values of \(m\) and \(n,\) assuming that \(a \neq b\) (six graphs in total). (a) \(m=1, n=1,2,3\) (b) \(m=2, n=2,3\) (c) \(m=3, n=3\)

Functions of the form $$f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} $$ arise in a wide variety of statistical problems. (a) Use the first derivative test to show that \(f\) has a relative maximum at \(x=0,\) and confirm this by using a graphing utility to graph \(f\) (b) Sketch the graph of $$f(x)=\frac{1}{\sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2} $$ where \(\mu\) is a constant, and label the coordinates of the relative extrema.

In each part, make a rough sketch of the graph using asymptotes and appropriate limits but no derivatives. Compare your sketch to that generated with a graphing utility. (a) \(y=\frac{3 x^{2}-8}{x^{2}-4}\) (b) \(y=\frac{x^{2}+2 x}{x^{2}-1}\) (c) \(y=\frac{2 x-x^{2}}{x^{2}+x-2}\) (d) \(y=\frac{x^{2}}{x^{2}-x-2}\)

(a) Use a CAS to graph the function $$f(x)=\frac{x^{4}+1}{x^{2}+1} $$ and use the graph to estimate the \(x\) -coordinates of the relative extrema. (b) Find the exact \(x\) -coordinates by using the CAS to solve the equation \(f^{\prime}(x)=0\).

Sketch the graph of $$y=\frac{1}{(x-a)(x-b)}$$ assuming that \(a \neq b\)

Suppose that the spread of a flu virus on a college campus is modeled by the function$$y(t)=\frac{1000}{1+999 e^{-0.9 t}}$$ where \(y(t)\) is the number of infected students at time \(t\) (in days, starting with \(t=0\) ). Use a graphing utility to estimate the day on which the virus is spreading most rapidly.

Consider the family of curves \(y=x e^{-b x}(b>0)\) (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(b\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

Let \(y=1 /\left(1+x^{2}\right) .\) Find the values of \(x\) for which \(y\) is increasing and decreasing most rapidly.

Let \(h\) and \(g\) have relative maxima at \(x_{0} .\) Prove or disprove: (a) \(h+g\) has a relative maximum at \(x_{0}\). (b) \(h-g\) has a relative maximum at \(x_{0}\).

Consider the family of curves \(y=e^{-b x^{2}}(b>0)\) (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(b\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

(a) Determine whether the following limits exist, and if so, find them: \(\lim _{x \rightarrow+\infty} e^{x} \cos x, \quad \lim _{x \rightarrow-\infty} e^{x} \cos x\) (b) Sketch the graphs of \(y=e^{x}, y=e^{-x},\) and \(y=e^{x} \cos x\) in the same coordinate system, and label any points of intersection. (c) Use a graphing utility to generate some members of the family \(y=e^{a x} \cos b x(a>0 \text { and } b>0),\) and discuss the effect of varying \(a\) and \(b\) on the shape of the curve.

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$\frac{x^{2}-2}{x}$$

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$\frac{x^{2}-2 x-3}{x+2}$$

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$\frac{(x-2)^{3}}{x^{2}}$$

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$\frac{4-x^{3}}{x^{2}}$$

Sketch the graph of the rational function. Show all vertical, horizontal, and oblique asymptotes. $$x+1-\frac{1}{x}-\frac{1}{x^{2}}$$

Find all values of \(x\) where the graph of $$y=\frac{2 x^{3}-3 x+4}{x^{2}}$$ crosses its oblique asymptote.

Let \(f(x)=\left(x^{3}+1\right) / x .\) Show that the graph of \(y=f(x)\) approaches the curve \(y=x^{2}\) "asymptotically" in the sense that \(\lim _{x \rightarrow+\infty}\left[f(x)-x^{2}\right]=0 \quad\) and \(\quad \lim _{x \rightarrow-\infty}\left[f(x)-x^{2}\right]=0\) Sketch the graph of \(y=f(x)\) showing this asymptotic behavior.

A rectangular plot of land is to be fenced off so that the area enclosed will be \(400 \mathrm{ft}^{2} .\) Let \(L\) be the length of fencing needed and \(x\) the length of one side of the rectangle. Show that \(L=2 x+800 / x\) for \(x>0,\) and sketch the graph of \(L\) versus \(x\) for \(x>0\)

A box with a square base and open top is to be made from sheet metal so that its volume is 500 in \(^{3} .\) Let \(S\) be the area of the surface of the box and \(x\) the length of a side of the square base. Show that \(S=x^{2}+2000 / x\) for \(x>0,\) and sketch the graph of \(S\) versus \(x\) for \(x>0\)