91Ó°ÊÓ

Finding Extrema and Points of Inflection In Exercises \(71-78\) , find the extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. $$ f(x)=x e^{-x} $$

Verifying an Integration Rule In Exercises \(75-77\) , verify the rule by differentiating. Let \(a>0\) . $$ \int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \arctan \frac{u}{a}+C $$

Finding an Indefinite Integral In Exercises \(71-78\) , find the indefinite integral. $$ \int(x+4) 6^{(x+4)^{2}} d x $$

Finding a Derivative Implicitly In Exercises \(73-76,\) use implicit differentiation to find \(d y / d x .\) $$ 4 x y+\ln x^{2} y=7 $$

In Exercises 75–82, find the indefinite integral using the formulas from Theorem 5.20. Exercises 75–82, find the indefinite integral using the formulas from Theorem 5.20. $$ \int \frac{1}{\sqrt{1+e^{2 x}}} d x $$

Verifying an Integration Rule In Exercises \(75-77\) , verify the rule by differentiating. Let \(a>0\) . $$ \int \frac{d u}{u \sqrt{u^{2}-a^{2}}}=\frac{1}{a} \operatorname{arcsec} \frac{|u|}{a}+C $$

Finding an Indefinite Integral In Exercises \(71-78\) , find the indefinite integral. $$ \int \frac{3^{2 x}}{1+3^{2 x}} d x $$

Finding Extrema and Points of Inflection In Exercises \(71-78\) , find the extrema and the points of inflection (if any exist) of the function. Use a graphing utility to graph the function and confirm your results. $$ g(t)=1+(2+t) e^{-t} $$

Numerical Integration In Exercises \(75-78\) , use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let \(n=4\) and round your answer to four decimal places. Use a graphing utility to verify your result. $$ \int_{2}^{6} \ln x d x $$

Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point. \(x^{2}+x \arctan y=y-1, \quad\left(-\frac{\pi}{4}, 1\right)\)