91Ó°ÊÓ

Proof Graph $$y_{1}=\frac{x}{1+x^{2}}, \quad y_{2}=\arctan x, \quad\( and \)\quad y_{3}=x$$ on \([0,10] .\) Prove that $$\frac{x}{1+x^{2}}<\arctan x0$$

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. $$ \int_{-\pi / 3}^{\pi / 3} \sec x d x $$

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. $$ f(x)=-2+e^{3 x}(4-2 x) $$

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

Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point. \(\arctan (x y)=\arcsin (x+y), \quad(0,0)\)

Exercises 75–82, find the indefinite integral using the formulas from Theorem 5.20. v$$ \int \frac{x}{9-x^{4}} d x $$

Use the functions \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the given function. \(g^{-1} \circ f^{-1}\)

In Exercises 79–84, locate any relative extrema and points of inflection. Use a graphing utility to confirm your results. $$ y=\frac{x^{2}}{2}-\ln x $$

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

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