91Ó°ÊÓ

Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Point }} \\ $y=37-\sec ^{3}(2 x) &\quad (0,36) \end{array} $$

Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Point }} \\ y=\frac{1}{x}+\sqrt{\cos x} &\quad\left(\frac{\pi}{2}, \frac{2}{\pi}\right) \end{array} $$

Find \(k\) such that the line is tangent to the graph of the function. \(\begin{array}{ll}\text { Function } & \text { Line } \\ \text { } f(x)=k \sqrt{x} \quad y=x+4\end{array}\)

(a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(f(x)=\frac{(x-1)}{(x+1)}, \quad\left(2, \frac{1}{3}\right)\)

Evaluate \(f(2)\) and \(f(2.1)\) and use the results to approximate \(f^{\prime}(2)\). \(f(x)=\frac{1}{4} x^{3}\)

Use a graphing utility to graph the function and its derivative in the same viewing window. Label the graphs and describe the relationship between them. \(f(x)=\frac{1}{\sqrt{x}}\)

(a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of the graphing utility to confirm your results. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Point }} \\ f(x)=\sqrt{3 x^{2}-2} &\quad (3,5) \end{array} $$

Sketch the graph of a function \(f\) such that \(f^{\prime}>0\) for all \(x\) and the rate of change of the function is decreasing.

Use a graphing utility to graph the function and its derivative in the same viewing window. Label the graphs and describe the relationship between them. \(f(x)=\frac{x^{3}}{4}-3 x\)

(a) find an equation of the tangent line to the graph of \(f\) at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of the graphing utility to confirm your results. $$ \begin{array}{ll} \underline{\text { Function }} & \underline{\text { Point }} \\ f(x)=\frac{1}{3} x \sqrt{x^{2}+5} &\quad (2,2) \end{array} $$