Chapter 7

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

Find the limit. $$ \lim _{x \rightarrow 3} \frac{\sqrt{x+1}}{x-4} $$

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

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

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. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 600 & 1200 & 1800 & 2400 & 3000 \\ \hline d R / d x & & & & & \\ \hline d P / d x & & & & & \\ \hline P & & & & & \\ \hline \end{array} $$

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{1}{x-2}\) \([1,4]\)

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

Find the limit. $$ \lim _{x \rightarrow 3} \frac{\sqrt{x+1}-1}{x} $$

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

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

Use the table to answer the questions below. $$ \begin{array}{|rc|rc|} \hline \begin{array}{c} \text { Quantity } \\ \text { produced } \\ \text { and sold } \\ (Q) \end{array} & \begin{array}{c} \text { Price } \\ (p) \end{array} & \begin{array}{c} \text { Total } \\ \text { revenue } \\ (T R) \end{array} & \begin{array}{c} \text { Marginal } \\ \text { revenue } \\ (M R) \end{array} \\ \hline 0 & 160 & 0 & \- \\ 2 & 140 & 280 & 130 \\ 4 & 120 & 480 & 90 \\ 6 & 100 & 600 & 50 \\ 8 & 80 & 640 & 10 \\ 10 & 60 & 600 & -30 \\ \hline \end{array} $$ (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 \(f^{\prime}(x)\). $$ f(x)=x^{2}+4 x+\frac{1}{x} $$

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{x}{x^{2}-4 x+3}\) \([0,4]\)

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

Find the limit. $$ \lim _{x \rightarrow 5} \frac{\sqrt{x+4}-2}{x} $$

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

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

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{x^{2}-16}{x-4}\)

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 limit. $$ \lim _{x \rightarrow 1} \frac{\frac{1}{x+4}-\frac{1}{4}}{x} $$

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

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

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{2 x^{2}+x}{x}\)

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 limit. $$ \lim _{x \rightarrow 2} \frac{\frac{1}{x+2}-\frac{1}{2}}{x} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=2\left(x^{2}-1\right)^{3} $$

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{x^{3}+x}{x}\)

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. $$ f(x)=(x-1)^{2} ;(-2,9) $$

find the limit $$ \lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=3(9 x-4)^{4} $$

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\frac{x-3}{4 x^{2}-12 x}\)

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. $$ f(x)=2 x^{2}-1 ;(0,-1) $$

find the limit $$ \lim _{x \rightarrow-1} \frac{2 x^{2}-x-3}{x+1} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\sqrt{4 x^{2}-7} $$

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.

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

Sketch the graph of the function and describe the interval(s) on which the function is continuous. \(f(x)=\left\\{\begin{array}{ll}x^{2}+1, & x<0 \\ x-1, & x \geq 0\end{array}\right.\)

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. $$ f(x)=\sqrt{x}+1 ;(4,3) $$

find the limit $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4 x+4} $$

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=x \sqrt{x^{2}+5} $$

The annual inventory cost for a manufacturer is given by \(C=1,008,000 / Q+6.3 Q\) where \(Q\) is the order size when the inventory is replenished. Find the change in annual cost when \(Q\) is increased from 350 to 351 , and compare this with the instantaneous rate of change when \(Q=350\).

find \(f^{\prime}(x)\). $$ f(x)=\frac{2 x^{2}-3 x+1}{x} $$