91Ó°ÊÓ

In Exercises \(1-8,\) complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 3} \frac{[1 /(x+1)]-(1 / 4)}{x-3}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 2.9 & 2.99 & 2.999 & 3.001 & 3.01 & 3.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} \end{aligned} $$

\( \text { Rate of Change } \) Each of the following is the slope of a line representing daily revenue \(y\) in terms of time \(x\) in days. Use the slope to interpret any change in daily revenue for a one-day increase in time. (a) \(m=400\) (b) \(m=100\) (c) \(m=0\)

Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of \(x\). (a) \(f(x)=\frac{8}{1+e^{-0.5 x}}\) (b) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\)

In Exercises 27-34, find the inverse function of \(f\). Graph (by hand) \(f\) and \(f^{-1}\). Describe the relationship between the graphs. $$ f(x)=2 x-3 $$

\( \text { Rate of Change } \) You are given the dollar value of a product in 2004 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value \(V\) of the product in terms of the year \(t\). (Let \(t=0\) represent 2000.) $$ \frac{2004 \text { Value }}{\$ 2540} $$ $$ \frac{\text { Rate }}{\$ 125 \text { increase per year }} $$

In Exercises \(35-40,\) find the inverse function of \(f\). Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=\sqrt[3]{x-1} $$

Use a graphing utility to graph the parabolas and find their points of intersection. Find an equation of the line through the points of intersection and graph the line in the same viewing window. $$ \begin{array}{l} y=x^{2}-4 x+3 \\ y=-x^{2}+2 x+3 \end{array} $$

Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).

Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on \(x\) in the denominator is greater than \(3 ?\) $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 0.5 & 0.2 & 0.1 & 0.01 & 0.001 & 0.0001 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ (a) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}\) (b) \(\lim _{x \rightarrow 0^{-}} \frac{x-\sin x}{x^{2}}\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}\) (d) \(\lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}\)

Show that \(f\) is one-to-one on the indicated interval and therefore has an inverse function on that interval. $$ f(x)=\cos x \quad[0, \pi] $$