91Ó°ÊÓ

Use a graphing utility to make rough estimates of the locations of all horizontal tangent lines, and then find their exact locations by differentiating. \(y=\frac{1}{3} x^{3}-\frac{3}{2} x^{2}+2 x\)

Find an equation for the tangent line to the graph at the specified value of \(x\). \(y=\frac{x}{\sqrt{1-x^{2}}}, x=0\)

Show that $$ f(x)=\left\\{\begin{array}{ll} x^{2} \sin (1 / x), & x \neq 0 \\ 0, & x=0 \end{array}\right. $$ is continuous and differentiable at \(x=0 .\) Sketch the graph of \(f\) near \(x=0\).

Suppose that \(f\) is a function that is differentiable everywhere. Explain the relationship, if any, between the periodicity of \(f\) and that of \(f^{\prime}\). That is, if \(f\) is periodic, must \(f^{\prime}\) also be periodic? If \(f^{\prime}\) is periodic, must \(f\) also be periodic?

Use a graphing utility to make rough estimates of the locations of all horizontal tangent lines, and then find their exact locations by differentiating. \(y=\frac{x^{2}+9}{x}\)

Suppose that a function \(f\) is differentiable at \(x_{0}\) and that \(f^{\prime}\left(x_{0}\right)>0 .\) Prove that there exists an open interval containing \(x_{0}\) such that if \(x_{1}\) and \(x_{2}\) are any two points in this interval with \(x_{1}

Find a function \(y=a x^{2}+b x+c\) whose graph has an \(x\) -intercept of 1 , a \(y\) -intercept of \(-2\), and a tangent line with a slope of \(-1\) at the \(y\) -intercept.

Find \(d^{2} y / d x^{2}\) \(y=x \cos (5 x)-\sin ^{2} x\)

Suppose that a function \(f\) is differentiable at \(x_{0}\) and define \(g(x)=f(m x+b)\), where \(m\) and \(b\) are constants. Prove that if \(x_{1}\) is a point at which \(m x_{1}+b=x_{0}\), then \(g(x)\) is differentiable at \(x_{1}\) and \(g^{\prime}\left(x_{1}\right)=m f^{\prime}\left(x_{0}\right)\).

Find \(d^{2} y / d x^{2}\) \(y=\sin \left(3 x^{2}\right)\)