91Ó°ÊÓ

Given the following table of values, find the indicated derivatives in parts (a) and (b). $$ \begin{array}{|c|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline 3 & 5 & -2 & 5 & 7 \\ \hline 5 & 3 & -1 & 12 & 4 \\ \hline \end{array} $$ (a) \(F^{\prime}(3)\), where \(F(x)=f(g(x))\) (b) \(G^{\prime}(3)\), where \(G(x)=g(f(x))\)

If a particle moves at constant velocity, what can you say about its position versus time curve?

Find \(f^{\prime}(x)\) $$ f(x)=\left(3 x^{2}+6\right)\left(2 x-\frac{1}{4}\right) $$

Find \(d y / d x\). $$ y=\pi^{3} $$

Find \(f^{\prime}(x)\) $$ f(x)=\left(2-x-3 x^{3}\right)\left(7+x^{5}\right) $$

Sketch the graph of a function \(f\) for which \(f(0)=0\), \(f^{\prime}(0)=0\), and \(f^{\prime}(x)>0\) if \(x<0\) or \(x>0 .\)

An automobile, initially at rest, begins to move along a straight track. The velocity increases steadily until suddenly the driver sees a concrete barrier in the road and applies the brakes sharply at time \(t_{0} .\) The car decelerates rapidly, but it is too late - the car crashes into the barrier at time \(t_{1}\) and instantaneously comes to rest. Sketch a position versus time curve that might represent the motion of the car. Indicate how characteristics of your curve correspond to the events of this scenario.

Given the following table of values, find the indicated derivatives in parts (a) and (b). $$ \begin{array}{|r|c|c|c|c|} \hline x & f(x) & f^{\prime}(x) & g(x) & g^{\prime}(x) \\ \hline-1 & 2 & 3 & 2 & -3 \\ \hline 2 & 0 & 4 & 1 & -5 \\ \hline \end{array} $$ (a) \(F^{\prime}(-1)\), where \(F(x)=f(g(x))\) (b) \(G^{\prime}(-1)\), where \(G(x)=g(f(x))\)

Find \(d y / d x\). $$ y=\sqrt{2} x+(1 / \sqrt{2}) $$

Find \(f^{\prime}(x)\). $$ f(x)=\frac{\sin x}{x^{2}+\sin x} $$