Chapter 2

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

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ \begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=(x-1)^{2}(x-2)} & {(0,-2)} \end{array}$$

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

Find the rate of change of \(x\) with respect to \(p .\) \(p=\frac{4}{0.000001 x^{2}+0.05 x+1} x \geq 0\)

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

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. $$ f(x)=3(9 x-4)^{4} $$

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

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.

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

Find the rate of change of \(x\) with respect to \(p .\) \(p=\sqrt{\frac{200-x}{2 x}}, \quad 0

A ball is propelled straight upward from ground level with an initial velocity of 144 feet per second. (a) Write the position, velocity, and acceleration functions of the ball. (b) When is the ball at its highest point? How high is this point? (c) How fast is the ball traveling when it hits the ground? How is this speed related to the initial velocity?

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. $$ f(x)=\sqrt{4 x^{2}-7} $$

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

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

Find the rate of change of \(x\) with respect to \(p .\) \(p=\sqrt{\frac{500-x}{2 x}}, \quad 0

A brick becomes dislodged from the top of the Empire State Building (at a height of 1250 feet) and falls to the sidewalk below. (a) Write the position, velocity, and acceleration functions of the brick. (b) How long does it take the brick to hit the sidewalk? (c) How fast is the brick traveling when it hits the sidewalk?

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ \begin{array}{ll}{\text { Function }} & {\text { Point }} \\ {f(x)=\frac{2 x+1}{x-1}} & {(2,5)} \end{array}$$

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. $$ f(x)=\sqrt{x+2} ;(7,3) $$

\(M A K E A D E C I S I O N: F U E L\) COST \(\quad\) A car is driven \(15,000\) miles a year and gets \(x\) miles per gallon. Assume that the average fuel cost is \(\$ 2.95\) per gallon. Find the annual cost of fuel \(C\) as a function of \(x\) and use this function to complete the table. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {10} & {15} & {20} & {25} & {30} & {35} & {40} \\ \hline C & {} & {} & {} & {} & {} & {} \\ \hline d C / d x & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array} $$ Who would benefit more from a 1 mile per gallon increase in fuel efficiency - the driver who gets 15 miles per gallon or the driver who gets 35 miles per gallon? Explain.

Find \(f^{\prime}(x)\) $$ f(x)=\frac{4 x^{3}-3 x^{2}+2 x+5}{x^{2}} $$

Velocity and Acceleration The velocity (in feet per second) of an automobile starting from rest is modeled by \(\frac{d s}{d t}=\frac{90 t}{t+10}\) Create a table showing the velocity and acceleration at 10 -second intervals during the first minute of travel. What can you conclude?

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ \begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {f(x)=\left(\frac{x+5}{x-1}\right)(2 x+1)} & {(0,-5)} \end{array}$$

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. $$ f(x)=\frac{1}{x} ;(1,1) $$

Find \(f^{\prime}(x)\) $$ f(x)=\frac{-6 x^{3}+3 x^{2}-2 x+1}{x} $$

Stopping Distance A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is given by \(s=-8.25 t^{2}+66 t,\) where \(s\) is measured in feet and \(t\) is measured in seconds. Create a table showing the position, velocity, and acceleration for each given value of \(t .\) What can you conclude?

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ \begin{array}{ll}{\text { Function }} & {\text { Point }} \\\ {g(x)=(x+2)\left(\frac{x-5}{x+1}\right)} & {(0,-10)} \end{array}$$

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. $$ f(x)=\frac{1}{x-1} ;(2,1) $$

Dow Jones Industrial Average The table shows the year-end closing prices \(p\) of the Dow Jones Industrial Average (DJIA) from 1992 through \(2006,\) where \(t\) is the year, and \(t=2\) corresponds to \(1992 .\) $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {2} & {3} & {4} & {5} & {6} \\ \hline p & {3301.11} & {3754.09} & {3834.44} & {5117.12} & {6448.26} \\\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {7} & {8} & {9} & {10} & {11} \\\ \hline p & {7908.24} & {9181.43} & {11,497.12} & {10,786.85} & {10,021.50} \\\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|c|}\hline t & {12} & {13} & {14} & {15} & {16} \\\ \hline p & {8341.63} & {10,453.92} & {10,783.01} & {10,717.50} & {12,463.15} \\\ \hline\end{array} $$ (a) Determine the average rate of change in the value of the DJIA from 1992 to 2006 . (b) Estimate the instantaneous rate of change in 1998 by finding the average rate of change from 1996 to 2000 . (c) Estimate the instantaneous rate of change in 1998 by finding the average rate of change from 1997 to 1999 . (d) Compare your answers for parts (b) and (c). Which interval do you think produced the best estimate for the instantaneous rate of change in \(1998 ?\)

Find \(f^{\prime}(x)\) $$ f(x)=x^{4 / 5}+x $$

The numbers (in thousands) of cases of HIV/AIDS reported in the years 2001 through 2005 can be modeled by \(y^{2}-1141.6=24.9099 t^{3}-183.045 t^{2}+452.79 t\) where \(t\) represents the year, with \(t=1\) corresponding to 2001.

In Exercises 47 and \(48,\) use a graphing utility to graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) in the same viewing window. What is the relationship among the degree of \(f\) and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives? $$ f(x)=x^{2}-6 x+6 $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\frac{\sqrt{x}+1}{x^{2}+1} $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ \begin{array}{ll}{\text { Function }} & {\text { Line }} \\\ {f(x)=-\frac{1}{4} x^{2}} & {x+y=0}\end{array} $$

Find the point(s), if any, at which the graph of has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x-1} $$

Find \(f^{\prime}(x)\) $$ f(x)=x^{1 / 3}-1 $$

In Exercises 47 and \(48,\) use a graphing utility to graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) in the same viewing window. What is the relationship among the degree of \(f\) and the degrees of its successive derivatives? In general, what is the relationship among the degree of a polynomial function and the degrees of its successive derivatives? $$ f(x)=3 x^{3}-9 x $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{2 x}{x+1}} $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ \begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{2}+1} & {2 x+y=0}\end{array} $$

Find the point(s), if any, at which the graph of has a horizontal tangent. $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

(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. $$ {y=-2 x^{4}+5 x^{2}-3} \quad (1,0) $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{\frac{x+1}{x}} $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ \begin{array}{ll}{\text { Function }} & {\text { Line }} \\\ {f(x)=-\frac{1}{2} x^{3}} & {6 x+y+4=0}\end{array} $$

Find the point(s), if any, at which the graph of has a horizontal tangent. $$ f(x)=\frac{x^{4}}{x^{3}+1} $$

(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. $$ y=x^{3}+x \quad(-1,-2) $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{x}\left(2-x^{2}\right) $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$ \begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{2}-x} & {x+2 y-6=0}\end{array} $$

Find the point(s), if any, at which the graph of has a horizontal tangent. $$f(x)=\frac{x^{4}+3}{x^{2}+1}$$

(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.. $$ f(x)=\sqrt[3]{x}+\sqrt[5]{x} \quad (1,2) $$