91Ó°ÊÓ

Use the position function to find the velocity at time \(t=t_{0} .\) Assume units of feet and seconds.$$s(t)=t^{2}-\sin 2 t, t_{0}=0$$

Find an equation of the tangent line to $$f(x)=x^{2} \ln x$$

Find all functions \(g\) such that \(g^{\prime}(x)=f(x).\) $$f(x)=x^{2}$$

Find the derivative of the given function. $$f(x)=\tan ^{-1} \sqrt{x}$$

The given function represents the height of an object. Compute the velocity and acceleration at time \(t=t_{0} .\) Is the object going up or down? Is the speed of the object increasing or decreasing? $$h(t)=-16 t^{2}+40 t+5, t_{0}=1$$

Find all functions \(g\) such that \(g^{\prime}(x)=f(x).\) $$f(x)=9 x^{4}$$

The given function represents the height of an object. Compute the velocity and acceleration at time \(t=t_{0} .\) Is the object going up or down? Is the speed of the object increasing or decreasing? $$h(t)=-16 t^{2}+40 t+5, t_{0}=2$$

Use the position function to find the velocity at time \(t=t_{0} .\) Assume units of feet and seconds. \(s(t)=t \cos \left(t^{2}+\pi\right), t_{0}=0\)

Find an equation of the tangent line to $$f(x)=2 \ln x^{3}$$

In exercise \(29,\) if the baseball has mass \(M\) kg at speed \(45 \mathrm{m} / \mathrm{s}\) and the bat has mass \(1.05 \mathrm{kg}\) at speed \(40 \mathrm{m} / \mathrm{s}\), the ball's initial speed is \(u(M)=\frac{86.625-45 M}{M+1.05} \mathrm{m} / \mathrm{s} .\) Compute \(u^{\prime}(M)\) and interpret its sign (positive or negative) in baseball terms.