Chapter 2

Population Growth The population \(P\) (in thousands) of Japan can be modeled by \(P=-14.71 t^{2}+785.5 t+117,216\) where \(t\) is time in years, with \(t=0\) corresponding to 1980 . (a) Evaluate \(P\) for \(t=0,10,15,20,\) and \(25 .\) Explain these values. (b) Determine the population growth rate, \(d P / d t\) (c) Evaluate \(d P / d t\) for the same values as in part (a). Explain your results.

Find the value of the derivative of the function at the given point. $$ f(x)=3(5-x)^{2} \quad(5,0) $$

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{3 x^{3}+4 x} $$

Use the limit definition to find the derivative of the function. $$ f(x)=\sqrt{x+2} $$

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

Health The temperature \(T\) (in degrees Fahrenheit) of a person during an illness can be modeled by the equation \(T=-0.0375 t^{2}+0.3 t+100.4,\) where \(t\) is time in hours since the person started to show signs of a fever. (a) Use a graphing utility to graph the function. Be sure to choose an appropriate window. (b) Do the slopes of the tangent lines appear to be positive or negative? What does this tell you? (c) Evaluate the function for \(t=0,4,8,\) and \(12 .\) (d) Find \(d T / d t\) and explain its meaning in this situation. (e) Evaluate \(d T / d t\) for \(t=0,4,8,\) and \(12 .\)

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}-\frac{4}{x}-3 x^{-2} $$

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(x^{2}+y^{2}=100 \quad(8,6)\) and \((-6,8)\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=x^{3}-9 x^{2}+27 x-27 $$

Use the General Power Rule to find the derivative of the function. $$ y=\sqrt[3]{9 x^{2}+4} $$

Use the limit definition to find the derivative of the function. $$ f(t)=t^{3}-12 t $$

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

Marginal Profit The profit \(P(\text { in dollars) from selling } x\) units of a product is given by $$P=36,000+2048 \sqrt{x}-\frac{1}{8 x^{2}}, \quad 150 \leq x \leq 275$$ Find the marginal profit for each of the following sales. $$ \begin{array}{ll}{\text { (a) } x=150} & {\text { (b) } x=175 \quad \text { (c) } x=200} \\ {\text { (d) } x=225} & {\text { (e) } x=250 \quad \text { (f) } x=275}\end{array} $$

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

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(x^{2}+y^{2}=9 \quad(0,3)\) and \((2, \sqrt{5})\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=3 x^{3}-9 x+1 $$

Use the General Power Rule to find the derivative of the function. $$ y=2 \sqrt{4-x^{2}} $$

Use the limit definition to find the derivative of the function. $$ f(t)=t^{3}+t^{2} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ h(t)=\frac{t+2}{t^{2}+5 t+6} $$

Profit The monthly demand function and cost function for \(x\) newspapers at a newsstand are given by \(p=5-0.001 x\) and \(C=35+1.5 x\) (a) Find the monthly revenue \(R\) as a function of \(x .\) (b) Find the monthly profit \(P\) as a function of \(x .\) (c) Complete the table. $$ \begin{array}{|l|c|c|c|c|c|}\hline x & {600} & {1200} & {1800} & {2400} & {3000} \\ \hline d R / d x & {} & {} & {} & {} \\ \hline d P / d x & {} & {} & {} & {} \\ \hline P & {} & {} & {} & {} \\ \hline\end{array} $$

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}-2 x-\frac{2}{x^{4}} $$

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(y^{2}=5 x^{3} \quad(1, \sqrt{5})\) and \((1,-\sqrt{5})\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=(x+3)(x-4)(x+5) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=-3 \sqrt[4]{2-9 x} $$

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

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ g(s)=\frac{s^{2}-2 s+5}{\sqrt{s}} $$

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?

Find \(f^{\prime}(x)\) $$ f(x)=x^{2}+4 x+\frac{1}{x} $$

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(4 x y+x^{2}=5 \quad(1,1)\) and \((5,-1)\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=(x+2)(x-2)(x+3)(x-3) $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(25+x^{2}\right)^{-1 / 2} $$

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

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

Marginal Profit When the price of a glass of lemonade at a lemonade stand was \(\$ 1.75,400\) glasses were sold. When the price was lowered to \(\$ 1.50,500\) glasses were sold. Assume that the demand function is linear and that the variable and fixed costs are \(\$ 0.10\) and \(\$ 25\), respectively. (a) Find the profit \(P\) as a function of \(x,\) the number of glasses of lemonade sold. (b) Use a graphing utility to graph \(P,\) and comment about the slopes of \(P\) when \(x=300\) and when \(x=700\). (c) Find the marginal profits when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.

Find \(f^{\prime}(x)\) $$ f(x)=x\left(x^{2}+1\right) $$

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(x^{3}+y^{3}=8 \quad(0,2)\) and \((2,0)\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=x \sqrt{x^{2}-1} $$

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

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=\frac{1}{2} x^{2} ;(2,2) $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ g(x)=\left(\frac{x-3}{x+4}\right)\left(x^{2}+2 x+1\right) $$

Marginal cost The cost \(C\) of producing \(x\) units is modeled by \(C=v(x)+k,\) where \(v\) represents the variable cost and \(k\) represents the fixed cost. Show that the marginal cost is independent of the fixed cost.

Find \(f^{\prime}(x)\) $$ f(x)=\left(x^{2}+2 x\right)(x+1) $$

Find equations of the tangent lines to the graph at the given points. Use a graphing utility to graph the equation and the tangent lines in the same viewing window. Equation \(\quad\) Points \(y^{2}=\frac{x^{3}}{4-x} \quad(2,2)\) and \((2,-2)\)

find the second derivative and solve the equation \(f^{\prime \prime}(x)=0\) $$ f(x)=x \sqrt{4-x^{2}} $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=(4-3 x)^{-5 / 2} $$

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=-x^{2} ;(-1,-1) $$

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

Marginal Profit When the admission price for a baseball game was \(\$ 6\) per ticket, \(36,000\) tickets were sold. When the price was raised to \(\$ 7,\) only \(33,000\) tickets were sold. Assume that the demand function is linear and that the variable and fixed costs for the ballpark owners are \(\$ 0.20\) and \(\$ 85,000,\) respectively. (a) Find the profit \(P\) as a function of \(x,\) the number of tickets sold. (b) Use a graphing utility to graph \(P,\) and comment about the slopes of \(P\) when \(x=18,000\) and when \(x=36,000\). (c) Find the marginal profits when \(18,000\) tickets are sold and when \(36,000\) tickets are sold.

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

Find the rate of change of \(x\) with respect to \(p .\) \(p=\frac{2}{0.00001 x^{3}+0.1 x} \quad x \geq 0\)