91Ó°ÊÓ

Slope Field In Exercises \(57-60\) , use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. $$ \begin{array}{l}{\frac{d y}{d x}=\frac{2 y}{\sqrt{16-x^{2}}}} \\\ {y(0)=2}\end{array} $$

Finding an Equation of a Tangent Line In Exercises \(55-62,\) find an equation of the tangent line to the graph of the function at the given point. $$ y=\ln \frac{e^{x}+e^{-x}}{2}, \quad(0,0) $$

In Exercises 41–64, find the derivative of the function. $$ f(x)=\ln \left(x+\sqrt{4+x^{2}}\right) $$

Using Technology to Find an Integral In Exercises \(57-62\) use a computer algebra system to find or evaluate the integral. $$ \int \frac{x^{2}}{x-1} d x $$

Slope Field In Exercises \(57-60\) , use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. $$ \begin{array}{l}{\frac{d y}{d x}=\frac{\sqrt{y}}{1+x^{2}}} \\\ {y(0)=4}\end{array} $$

Find an equation of the tangent line to the graph of the function at the given point. \(y=\frac{1}{2} \arccos x, \quad\left(-\frac{\sqrt{2}}{2}, \frac{3 \pi}{8}\right)\)

In Exercises 55–60, evaluate the integral. $$ \int_{0}^{\ln 2} 2 e^{-x} \cosh x d x $$

Finding an Equation of a Tangent Line In Exercises \(55-62,\) find an equation of the tangent line to the graph of the function at the given point. $$ y=x^{2} e^{x}-2 x e^{x}+2 e^{x}, \quad(1, e) $$

Find an equation of the tangent line to the graph of the function at the given point. \(y=\arctan \frac{x}{2}, \quad\left(2, \frac{\pi}{4}\right)\)

Discuss several ways in which the hyperbolic functions are similar to the trigonometric functions.