Chapter 2

Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=-6 x^{3}+8 x^{2}+200 x $$

Sales The profit for a product is increasing at a rate of \(\$ 5600\) per week. The demand and cost functions for the product are given by \(p=6000-25 x\) and \(C=2400 x+5200\). Find the rate of change of sales with respect to time when the weekly sales are \(x=44\) units.

Use the General Power Rule to find the derivative of the function. $$ g(x)=(4-2 x)^{3} $$

find the given value. $$ f(x)=\sqrt{4-x} \quad f^{\prime \prime \prime}(-5) $$

Use the limit definition to find the derivative of the function. $$ f(x)=3 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\left(x^{3}-3 x\right)\left(2 x^{2}+3 x+5\right) $$

Find the marginal revenue for producing units. (The revenue is measured in dollars.) $$ R=50\left(20 x-x^{3 / 2}\right) $$

cost The annual cost (in millions of dollars) for a government agency to seize \(p \%\) of an illegal drug is given by \(C=\frac{528 p}{100-p}, \quad 0 \leq p<100\) The agency's goal is to increase \(p\) by \(5 \%\) per year. Find the rates of change of the cost when (a) \(p=30 \%\) and (b) \(p=60 \%\). Use a graphing utility to graph \(C .\) What happens to the graph of \(C\) as \(p\) approaches \(100 ?\)

$$ h(t)=\left(1-t^{2}\right)^{4} $$

Use the limit definition to find the derivative of the function. $$ f(x)=-2 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ h(t)=\left(t^{5}-1\right)\left(4 t^{2}-7 t-3\right) $$

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-2 x^{2}+72 x-145 $$

Use Example 6 as a model to find the derivative. $$ y=\frac{\sqrt{x}}{x} $$

Use the General Power Rule to find the derivative of the function. $$ h(x)=\left(6 x-x^{3}\right)^{2} $$

find the given value. $$ f(x)=x^{2}\left(3 x^{2}+3 x-4\right) \quad f^{\prime \prime \prime}(-2) $$

Use the limit definition to find the derivative of the function. $$ f(x)=-5 x $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ g(t)=\left(2 t^{3}-1\right)^{2} $$

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-0.25 x^{2}+2000 x-1,250,000 $$

find the given value. $$ g(x)=2 x^{3}\left(x^{2}-5 x+4\right) \quad g^{\prime \prime \prime}(0) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(4 x-x^{2}\right)^{3} $$

Use the limit definition to find the derivative of the function. $$ f(x)=4 x+1 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ h(p)=\left(p^{3}-2\right)^{2} $$

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-0.00025 x^{2}+12.2 x-25,000 $$

Find the value of the derivative of the function at the given point. $$ {f(x)=\frac{1}{x}}\quad {(1,1)} $$

find the higher-order derivative. $$ f^{\prime}(x)=2 x^{2} \quad f^{\prime \prime}(x) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(x^{2}-9\right)^{2 / 3} $$

Use the limit definition to find the derivative of the function. $$ g(s)=\frac{1}{3} s+2 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\sqrt[3]{x}(\sqrt{x}+3) $$

Find the marginal profit for producing units. (The profit is measured in dollars.) $$ P=-0.5 x^{3}+30 x^{2}-164.25 x-1000 $$

Find the value of the derivative of the function at the given point. $$ f(t)=4-\frac{4}{3 t} \quad\left(\frac{1}{2}, \frac{4}{3}\right) $$

find the higher-order derivative. $$ f^{\prime \prime}(x)=20 x^{3}-36 x^{2} \quad f^{\prime \prime \prime}(x) $$

Use the General Power Rule to find the derivative of the function. $$ f(t)=(9 t+2)^{2 / 3} $$

Use the limit definition to find the derivative of the function. $$ h(t)=6-\frac{1}{2} t $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\sqrt[3]{x}(x+1) $$

Marginal cost The cost \(C\) (in dollars) of producing \(x\) units of a product is given by \(C=3.6 \sqrt{x}+500 .\) (a) Find the additional cost when the production increases from 9 to 10 units. (b) Find the marginal cost when \(x=9 .\) (c) Compare the results of parts (a) and (b).

Find the value of the derivative of the function at the given point. $$ f(x)=-\frac{1}{2} x\left(1+x^{2}\right) \quad(1,-1) $$

find the higher-order derivative. $$ f^{\prime \prime \prime}(x)=(3 x-1) / x \quad f^{(4)}(x) $$

Use the General Power Rule to find the derivative of the function. $$ f(t)=\sqrt{t+1} $$

Use the limit definition to find the derivative of the function. $$ f(x)=x^{2}-4 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\frac{3 x-2}{2 x-3} $$

Marginal Revenue The revenue \(R\) (in dollars) from renting \(x\) apartments can be modeled by \(R=2 x\left(900+32 x-x^{2}\right)\) (a) Find the additional revenue when the number of rentals (a) Find the additional revenue when the number of rentals is increased from 14 to 15 . (b) Find the marginal revenue when \(x=14\). (c) Compare the results of parts (a) and (b).

Find the value of the derivative of the function at the given point. $$ {y=3 x\left(x^{2}-\frac{2}{x}\right)} \quad (2,18) $$

Use the General Power Rule to find the derivative of the function. $$ g(x)=\sqrt{5-3 x} $$

Use the limit definition to find the derivative of the function. $$ f(x)=1-x^{2} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\frac{x^{3}+3 x+2}{x^{2}-1} $$

Marginal Profit The profit \(P\) (in dollars) from selling \(x\) units of calculus textbooks is given by \(P=-0.05 x^{2}+20 x-1000\) (a) Find the additional profit when the sales increase from 150 to 151 units. (b) Find the marginal profit when \(x=150\). (c) Compare the results of parts (a) and (b).

Find the value of the derivative of the function at the given point. $$ y=(2 x+1)^{2} \quad(0,1) $$

Use the General Power Rule to find the derivative of the function. $$ s(t)=\sqrt{2 t^{2}+5 t+2} $$

Use the limit definition to find the derivative of the function. $$ h(t)=\sqrt{t-1} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\frac{3-2 x-x^{2}}{x^{2}-1} $$