91Ó°ÊÓ

For the following exercises, the volume \(V\) of a sphere with respect to its radius \(r\) is given by \(V=\frac{4}{3} \pi r^{3}\). Find the instantaneous rate of change of \(V\) when \(r=3 \mathrm{cm}\).

For the following exercises, the revenue generated by selling \(x\) items is given by \(R(x)=2 x^{2}+10 x\). Find the average change of the revenue function as \(x\) changes from \(x=10\) to \(x=20\).

For the following exercises, the revenue generated by selling \(x\) items is given by \(R(x)=2 x^{2}+10 x\). Find \(R^{\prime}(10)\) and interpret.

For the following exercises, the revenue generated by selling \(x\) items is given by \(R(x)=2 x^{2}+10 x\). Find \(R^{\prime}(15)\) and interpret. Compare \(R^{\prime}(15)\) to \(R^{\prime}(10),\) and explain the difference.

For the following exercises, the cost of producing \(x\) cellphones is described by the function \(C(x)=x^{2}-4 x+1000\). Find the average rate of change in the total cost as \(x\) changes from \(x=10\) to \(x=15\).

For the following exercises, the cost of producing \(x\) cellphones is described by the function \(C(x)=x^{2}-4 x+1000\). Find the approximate marginal cost, when 15 cellphones have been produced, of producing the \(16^{\text { th }}\) cellphone.

For the following exercises, the cost of producing \(x\) cellphones is described by the function \(C(x)=x^{2}-4 x+1000\). Find the approximate marginal cost, when 20 cellphones have been produced, of produced, of producing the \(21^{\text { st }}\) cellphone.

For the following exercises, use the definition for the derivative at a point \(x=a, \quad \lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\), to find the derivative of the functions. $$f(x)=\frac{1}{x^{2}}$$

For the following exercises, use the definition for the derivative at a point \(x=a, \quad \lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\), to find the derivative of the functions. $$f(x)=5 x^{2}-x+4$$

For the following exercises, use the definition for the derivative at a point \(x=a, \quad \lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}\), to find the derivative of the functions. $$f(x)=-x^{2}+4 x+7$$