91Ó°ÊÓ

Analyzing a Graph Using Technology In Exercises \(75-82,\) use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\frac{2 x}{\sqrt{3 x^{2}+1}} $$

Prove that if \(f\) is differentiable on \((-\infty, \infty)\) and \(f^{\prime}(x)<1\) for all real numbers, then \(f\) has at most one fixed point. A fixed point of a function \(f\) is a real number \(c\) such that \(f(c)=c .\)

Comparing Functions In Exercises 83 and \(84,\) (a) use a graphing utility to graph \(f\) and \(g\) in the same viewing window, (b) verify algebraically that \(f\) and \(g\) represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) $$ \begin{array}{l}{f(x)=-\frac{x^{3}-2 x^{2}+2}{2 x^{2}}} \\ {g(x)=-\frac{1}{2} x+1-\frac{1}{x^{2}}}\end{array} $$

The efficiency of an internal combustion engine is Efficiency $$(\%)=100\left[1-\frac{1}{\left(v_{1} / v_{2}\right)^{c}}\right]$$ where \(v_{1} / v_{2}\) is the ratio of the uncompressed gas to the compressed gas and \(c\) is a positive constant dependent on the engine design. Find the engine design. Find the limit of the efficiency as the compression ratio approaches infinity.

Modeling Data A heat probe is attached to the heat exchanger of a heating system. The temperature \(T\) (in degrees Celsius) is recorded \(t\) seconds after the furnace is started. The results for the first 2 minutes are recorded in the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {0} & {15} & {30} & {45} & {60} \\\ \hline T & {25.2^{\circ}} & {36.9^{\circ}} & {45.5^{\circ}} & {51.4^{\circ}} & {56.0^{\circ}} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline t & {75} & {90} & {105} & {120} \\ \hline T & {59.6^{\circ}} & {62.0^{\circ}} & {64.0^{\circ}} & {65.2^{\circ}} \\\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find a model of the form \(T_{1}=a t^{2}+b t+c\) for the data. (b) Use a graphing utility to graph \(T_{1} .\) (c) A rational model for the data is $$T_{2}=\frac{1451+86 t}{58+t}$$ Use a graphing utility to graph \(T_{2}\) (d) Find \(T_{1}(0)\) and \(T_{2}(0)\) (e) Find \(\lim _{t \rightarrow \infty} T_{2}\) . (f) Interpret the result in part (e) in the context of the problem. Is it possible to do this type of analysis using \(T_{1} ?\) Explain.