91Ó°ÊÓ

Given that \(g(5)=-3, \quad g^{\prime}(5)=6, \quad h(5)=3,\) and \(h^{\prime}(5)=-2,\) find \(f^{\prime}(5)\) (if possible) for each of the following. If it is not possible, state what additional information is required. (a) \(f(x)=g(x) h(x)\) (b) \(f(x)=g(h(x))\) (c) \(f(x)=\frac{g(x)}{h(x)}\) (d) \(f(x)=[g(x)]^{3}\)

The normal daily maximum temperatures \(T\) (in degrees Fahrenheit) for Denver, Colorado, are shown in the table. (Source: National Oceanic and Atmospheric Administration). $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Month } & \text { Jan } & \text { Feb } & \text { Mar } & \text { Apr } & \text { May } & \text { Jun } \\ \hline \text { Temperature } & 43.2 & 47.2 & 53.7 & 60.9 & 70.5 & 82.1 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Month } & \text { Jul } & \text { Aug } & \text { Sep } & \text { Oct } & \text { Nov } & \text { Dec } \\ \hline \text { Temperature } & 88.0 & 86.0 & 77.4 & 66.0 & 51.5 & 44.1 \\ \hline \end{array} \end{aligned} $$(a) Use a graphing utility to plot the data and find a model for the data of the form \(T(t)=a+b \sin (\pi t / 6-c)\) where \(T\) is the temperature and \(t\) is the time in months, with \(t=1\) corresponding to January. (b) Use a graphing utility to graph the model. How well does the model fit the data? (c) Find \(T^{\prime}\) and use a graphing utility to graph the derivative. (d) Based on the graph of the derivative, during what times does the temperature change most rapidly? Most slowly? Do your answers agree with your observations of the temperature changes? Explain.

Let \(u\) be a differentiable function of \(x\). Use the fact that \(|u|=\sqrt{u^{2}}\) to prove that \(\frac{d}{d x}[|u|]=u^{\prime} \frac{u}{|u|}, \quad u \neq 0\).

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ \begin{array}{l} f(x)=x \ln x \\ a=1 \end{array} $$