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Chapter 2: Problem 73
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is differentiable at a point, then it is continuous at that point.
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Use the table to answer the questions below. $$ \begin{array}{|cc|cc|}\hline \text { Quantity } & {} & {} & {} \\ {\text { produced }} & {} & {\text { Total }} & {\text { Marginal }} \\ {\text { and sold }} & {\text { Price }} & {(T R)} & {(M R)} \\ {(Q)} & {(p)} & {} & {(M R)} \\ \hline 0 & {160} & {0} & {-} \\ {2} & {140} & {280} & {130} \\ {4} & {120} & {480} & {90} \\ {6} & {100} & {600} & {50} \\ {8} & {80} & {640} & {10} \\ {10} & {60} & {600} & {-30} \\ \hline\end{array} $$ (a) Use the regression feature of a graphing utility to find a quadratic model that relates the total revenue \((T R)\) to the quantity produced and sold \((Q) .\) (b) Using derivatives, find a model for marginal revenue from the model you found in part (a). (c) Calculate the marginal revenue for all values of \(Q\) using your model in part (b), and compare these values with the actual values given. How good is your model?
Profit The demand function for a product is given by \(p=50 / \sqrt{x}\) for \(1 \leq x \leq 8000,\) and the cost function is given by \(C=0.5 x+500\) for \(0 \leq x \leq 8000\). Find the marginal profits for (a) \(x=900,\) (b) \(x=1600,\) (c) \(x=2500,\) and (d) \(x=3600\). If you were in charge of setting the price for this product, what price would you set? Explain your reasoning.
Use Example 6 as a model to find the derivative. $$ y=\frac{2}{3 x^{2}} $$
Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ h(x)=2-x ;[0,2] $$
(a)Find an equation of the tangent line to the graph of the function at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.. $$ f(x)=\sqrt[3]{x}+\sqrt[5]{x} \quad (1,2) $$
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