91Ó°ÊÓ

In Exercises \(5-12,\) approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than \(0.001 .\) Then find the zero(s) using a graphing utility and compare the results. $$ f(x)=x-e^{-x} $$

Use the Quotient Rule to differentiate the function. $$ f(x)=\frac{x}{x^{2}+1} $$

Find \(d y / d x\) by implicit differentiation. $$ e^{x y}+x^{2}-y^{2}=10 $$

In Exercises 5-8, show that the slopes of the graphs of \(f\) and \(f^{-1}\) are reciprocals at the indicated points. $$\begin{array}{ll}\text { Function } & \text { Point } \\\\\hline f(x)=\frac{4}{1+x^{2}}, \quad x \geq 0 & (1,2)\\\f^{-1}(x)=\sqrt{\frac{4-x}{x}} & (2,1)\end{array}$$

A point is moving along the graph of the given function such that \(d x / d t\) is 2 centimeters per second. Find \(d y / d t\) for the given values of \(x\). $$ y=\sin x \quad \text { (a) } x=\frac{\pi}{6} \quad \text { (b) } x=\frac{\pi}{4} \quad \text { (c) } x=\frac{\pi}{3} $$

Find the slope of the tangent line to the graph of the function at the given point. \(g(x)=5-x^{2}, \quad(2,1)\)

Find the derivative of the function. $$ y=t^{2}+2 t-3 $$

In Exercises \(5-12,\) approximate the zero(s) of the function. Use Newton's Method and continue the process until two successive approximations differ by less than \(0.001 .\) Then find the zero(s) using a graphing utility and compare the results. $$ f(x)=x-3+\ln x $$

Use the Quotient Rule to differentiate the function. $$ g(t)=\frac{t^{2}+2}{2 t-7} $$

Find the derivative of the function. $$ g(x)=x^{2}+4 x^{3} $$