91Ó°ÊÓ

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 that \(a, b, c\), and \(k\) are constants. $$ f(x)=\frac{a x+b}{c x+k} $$

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.

Paris, France, has a latitude of approximately \(49^{\circ} \mathrm{N}\). If \(t\) is the number of days since the start of 2009 , the number of hours of daylight in Paris can be approximated by $$ D(t)=4 \cos \left(\frac{2 \pi}{365}(t-172)\right)+12 $$ (a) Find \(D(40)\) and \(D^{\prime}(40) .\) Explain what this tells about daylight in Paris. (b) Find \(D(172)\) and \(D^{\prime}(172)\). Explain what this tells about daylight in Paris.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=\left(a x^{2}+b\right)^{3} $$

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=a x^{2}+b x+c$$

Find the derivative. Assume \(a, b, c, k\) are constants. $$Q=a P^{2}+b P^{3}$$

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.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=a x e^{-b x} $$

Find the derivative. Assume \(a, b, c, k\) are constants. $$v=a t^{2}+\frac{b}{t^{2}}$$