91Ó°ÊÓ

Find the equation of the tangent line to the graph of \(f(x)=\frac{2 x-5}{x+1}\) at the point at which \(x=0.\)

If you invest \(P\) dollars in a bank account at an annual interest rate of \(r \%,\) then after \(t\) years you will have \(B\) dollars, where $$B=P\left(1+\frac{r}{100}\right)^{t}$$ (a) Find \(d B / d t,\) assuming \(P\) and \(r\) are constant. In terms of money, what does \(d B / d t\) represent? (b) Find \(\overline{d B} / d r,\) assuming \(P\) and \(t\) are constant. In terms of money, what does \(d B / d r\) represent?

For \(f(t)=4-2 e^{t},\) find \(f^{\prime}(-1), f^{\prime}(0),\) and \(f^{\prime}(1) .\) Graph \(f(t),\) and draw tangent lines at \(t=-1, t=0,\) and \(t=1 .\) Do the slopes of the lines match the derivatives you found?

Find the derivative. Assume \(a, b, c, k\) are constants. $$V=\frac{4}{3} \pi r^{2} b$$

Find the equation of the tangent line to the graph of \(f(x)=x^{2} e^{-x}\) at \(x=0 .\) Check by graphing this function and the tangent line on the same axes.

The quantity of a drug, \(Q\) mg, present in the body \(t\) hours after an injection of the drug is given is $$Q=f(t)=100 t e^{-0.5 t}$$ Find \(f(1), f^{\prime}(1), f(5),\) and \(f^{\prime}(5) .\) Give units and interpret the answers.

Find the equation of the tangent line to the graph of \(y=3^{x}\) at \(x=1 .\) Check your work by sketching a graph of the function and the tangent line on the same axes.

A firm estimates that the total revenue, \(R\), received from the sale of \(q\) goods is given by $$R=\ln \left(1+1000 q^{2}\right)$$ Calculate the marginal revenue when \(q=10\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$P=a+b \sqrt{t}$$

Find the equation of the tangent line to \(y=e^{-2 t}\) at \(t=0 .\) Check by sketching the graphs of \(y=e^{-2 t}\) and the tangent line on the same axes.