Chapter 4

Use the zero or root feature of a graphing utility to approximate the solution of the logarithmic equation. $$\ln x^{2}-e^{x}=-3-\ln x^{2}$$

Let \(f(x)=\ln x\) and \(g(x)=x^{1 / n}\). (a) Use a graphing utility to graph \(g\) (for \(n=2\) ) and \(f\) in the same viewing window. (b) Determine which function is increasing at a greater rate as \(x\) approaches infinity. (c) Repeat parts (a) and (b) for \(n=3,4,\) and \(5 .\) What do you notice?

Simplify the expression.$$x y\left(x^{-1}+y^{-1}\right)^{-1}$$

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{aligned}&y_{1}=7\\\&y_{2}=2^{x-1}-5\end{aligned}$$

(a) Use a graphing utility to compare the graph of the function \(y=\ln x\) with the graph of each function. $$\begin{array}{l}y_{1}=x-1, y_{2}=(x-1)-\frac{1}{2}(x-1)^{2} , \\\y_{3}=(x-1)-\frac{1}{2}(x-1)^{2}+\frac{1}{3}(x-1)^{3} \end{array}$$ (b) Identify the pattern of successive polynomials given in part (a). Extend the pattern one more term and compare the graph of the resulting polynomial function with the graph of \(y=\ln x .\) What do you think the pattern implies?

Simplify the expression.$$x y\left(x^{-1}+y^{-1}\right)^{-1}$$

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=4 \\\y_{2}=3^{x+1}-2\end{array}$$

Simplify the expression.$$\left(\frac{2 x^{3}}{3 y}\right)^{-3}$$

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{aligned}&y_{1}=80\\\&y_{2}=4 e^{-0.2 x}\end{aligned}$$

(a) Use a graphing utility to complete the table for the function \(f(x)=(\ln x) / x.\) $$\begin{array}{|l|l|l|l|l|l|l|}\hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\\\\hline f(x) & & & & & & \\\\\hline\end{array}$$ (b) Use the table in part (a) to determine what value \(f(x)\) approaches as \(x\) increases without bound. Use the graphing utility to confirm your result.

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=500 \\\y_{2}=1500 e^{-x / 2}\end{array}$$

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{array}{l}y_{1}=3.25 \\\y_{2}=\frac{1}{2} \ln (x+2)\end{array}$$

Factor the polynomial. $$x^{2}+2 x-3$$

Use a graphing utility to approximate the point of intersection of the graphs. Round your result to three decimal places. $$\begin{aligned}&y_{1}=1.05\\\&y_{2}=\ln \sqrt{x-2}\end{aligned}$$

Factor the polynomial. $$x^{2}+4 x-5$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x^{2} e^{2 x}+2 x e^{2 x}=0$$

Factor the polynomial. $$12 x^{2}+5 x-3$$

Factor the polynomial. $$16 x^{2}-54 x-7$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$

Factor the polynomial. $$16 x^{2}-25$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$e^{-2 x}-2 x e^{-2 x}=0$$

Factor the polynomial. $$36 x^{2}-49$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$2 x \ln x+x=0$$

Factor the polynomial. $$2 x^{3}+x^{2}-45 x$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1-\ln x}{x^{2}}=0$$

Factor the polynomial. $$3 x^{3}-5 x^{2}-12 x$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$\frac{1+\ln x}{2}=0$$

Evaluate the function for \(f(x)=3 x+2\) and \(g(x)=x^{3}-1.\) $$(f+g)(2)$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$3 x \ln \left(\frac{1}{x}\right)-x=0$$

Evaluate the function for \(f(x)=3 x+2\) and \(g(x)=x^{3}-1.\) $$(f-g)(-1)$$

Solve the equation graphically. $$5 x-7=7+5 x$$

Solve the equation graphically. $$-2 x+3=8 x$$

Solve the equation graphically. $$\sqrt{3 x-2}=9$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two decimal places. $$r=7 \%$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two decimal places. $$r=6 \%$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two $$r=2.5 \%$$

Find the time required for a \(\$ 1000\) investment to (a) double at interest rate \(r,\) compounded continuously, and (b) triple at interest rate \(r\), compounded continuously. Round your results to two $$r=3.75 \%$$

The percent \(p\) (in decimal form) of the United States population who own a smartphone is given by $$p=\frac{1}{1+e^{-(t-93) / 22.5}}$$ where \(t\) is the number of months after smartphones were available on the market. Find the number of months \(t\) when the percent of the population owning smartphones is (a) \(50 \%\) and (b) \(80 \%\).

The percent \(m\) of American males between the ages of 18 and 24 who are no more than \(x\) inches tall is modeled by $$m(x)=\frac{100}{1+e^{-0.6114(x-69.71)}}$$ and the percent \(f\) of American females between the ages of 18 and 24 who are no more than \(x\) inches tall is modeled by $$f(x)=\frac{100}{1+e^{-0.66607(x-64.51)}}$$ (Source: U.S. National Center for Health Statistics) (a) Use a graphing utility to graph the two functions in the same viewing window. (b) Use the graphs in part (a) to determine the horizontal asymptotes of the functions. Interpret their meanings in the context of the problem. (c) What is the average height for each sex?

The numbers \(y\) of commercial banks in the United States from 2007 through 2013 can be modeled by $$y=11,912-2340.1 \ln t, \quad 7 \leq t \leq 13$$ where \(t\) represents the year, with \(t=7\) corresponding to \(2007 .\) In what year were there about 6300 commercial banks? (Source: Federal Deposit Insurance Corp.)

The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}\) (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet.

The table shows the numbers \(N\) of college-bound seniors intending to major in engineering who took the SAT exam from 2008 through \(2013 .\) The data can be modeled by the logarithmic function $$N=-152,656+111,959.9 \ln t$$ where \(t\) represents the year, with \(t=8\) corresponding to 2008 . (Source: The College Board) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number } N \\\\\hline 2008 & 81,338 \\\2009 & 88,719 \\\2010 & 108,389 \\\2011 & 116,746 \\\2012 & 127,061 \\\2013 & 132,275 \\\\\hline\end{array}$$ (a) According to the model, in what year would 150,537 seniors intending to major in engineering take the SAT exam? (b) Use a graphing utility to graph the model with the data, and use the graph to verify your answer in part (a). (c) Do you think this is a good model for predicting future values? Explain.

An exponential equation must have at least one solution.

A logarithmic equation can have at most one extraneous solution.

Let \(f(x)=\log _{a} x\) and \(g(x)=a^{x},\) where \(a>1.\) (a) Let \(a=1.2\) and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of \(a\) for which the two graphs have one point of intersection. (c) Determine the value(s) of \(a\) for which the two graphs have two points of intersection.

Is the time required for a continuously compounded investment to quadruple twice as long as the time required for it to double? Give a reason for your answer and verify your answer algebraically.

Sketch the graph of the function. $$f(x)=3 x^{3}-4$$

Sketch the graph of the function. $$f(x)=|x-2|-8$$

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}2 x+1, & x<0 \\\\-x^{2}, & x \geq 0\end{array}\right.$$

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{l}x-9, x \leq 1 \\\x^{2}+1, x>1\end{array}\right.$$