Chapter 2

Modeling Data The table shows the retail values \(y\) (in billions of dollars) of motor homes sold in the United States for 2000 to \(2005,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} \\\ \hline y & {9.5} & {8.6} & {11.0} & {12.1} & {14.7} & {14.4} \\\ \hline\end{array} $$ (a) Use a graphing utility to find a cubic model for the total retail value of the motor homes. (b) Use a graphing utility to graph the model and plot the data in the same viewing window. How well does the model fit the data? (c) Find the first and second derivatives of the function. (d) Show that the retail value of motor homes was increasing from 2001 to 2004. (e) Find the year when the retail value was increasing at the greatest rate by solving \(y^{\prime \prime}(t)=0\) (f) Explain the relationship among your answers for parts (c), (d), and (e).

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=|x+3| $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x(x+1) $$

(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. $$ f(x)=\frac{1}{\sqrt[3]{x^{2}}}-x \quad(-1,2) $$

An object is thrown upward from the top of a 64-foot building with an initial velocity of 48 feet per second. (a) Write the position, velocity, and acceleration functions of the object. (b) When will the object hit the ground? (c) When is the velocity of the object zero? (d) How high does the object (e) Use a graphing utility to graph the position, velocity, and acceleration functions in the same viewing window. Write a short paragraph that describes the relationship among these functions.

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\left|x^{2}-9\right| $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x^{2}(x+1) $$

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=-x^{4}+3 x^{2}-1 $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$y=-\frac{4}{(t+2)^{2}}$$

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=(x-3)^{2 / 3} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x(x+1)(x-1) $$

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=x^{3}+3 x^{2} $$

determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(x+1)(x+2)(x+3)(x+4), \text { then } \frac{d^{5} y}{d x^{5}}=0 $$

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=x^{2 / 5} $$

Use a graphing utility to graph \(f\) and \(f^{\prime}\) on the interval \([-2,2] .\) $$ f(x)=x^{2}(x+1)(x-1) $$

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=\frac{1}{2} x^{2}+5 x $$

determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f^{\prime}(c)\) and \(g^{\prime}(c)\) are zero and \(h(x)=f(x) g(x),\) then \(h^{\prime}(c)=0\)

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\sqrt{x-1} $$

Use the demand function to find the rate of change in the demand \(x\) for the given price \(p .\) $$ x=275\left(1-\frac{3 p}{5 p+1}\right), p=\$ 4 $$

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. $$ y=x^{2}+2 x $$

determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The second derivative represents the rate of change of the first derivative.

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\frac{1}{\sqrt{x+2}} $$

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\frac{x^{2}}{x^{2}-4} $$

Use the demand function to find the rate of change in the demand \(x\) for the given price \(p .\) $$ x=300-p-\frac{2 p}{p+1}, p=\$ 3 $$

(a) sketch the graphs of \(f\) and \(g,(b)\) find \(f^{\prime}(1)\) and \(g^{\prime}(1),(c)\) sketch the tangent line to each graph when \(x=1,\) and \((d)\) explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). $$ \begin{array}{l}{f(x)=x^{3}} \\ {g(x)=x^{3}+3}\end{array} $$

Finding a Pattern Develop a general rule for \([x f(x)]^{(n)}\) where \(f\) is a differentiable function of \(x .\)

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\left\\{\begin{array}{ll}{x^{3}+3,} & {x<0} \\ {x^{3}-3,} & {x \geq 0}\end{array}\right. $$

Environment The model $$f(t)=\frac{t^{2}-t+1}{t^{2}+1}$$ measures the level of oxygen in a pond, where \(t\) is the time (in weeks) after organic waste is dumped into the pond. Find the rates of change of \(f\) with respect to \(t\) when (a) \(t=0.5,\) (b) \(t=2,\) and \((\mathrm{c}) t=8\)

(a) sketch the graphs of \(f\) and \(g,(b)\) find \(f^{\prime}(1)\) and \(g^{\prime}(1),(c)\) sketch the tangent line to each graph when \(x=1,\) and \((d)\) explain the relationship between \(f^{\prime}\) and \(g^{\prime}\). $$ \begin{array}{l}{f(x)=x^{2}} \\ {g(x)=3 x^{2}}\end{array} $$

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

Describe the \(x\) -values at which the function is differentiable. Explain your reasoning. $$ y=\left\\{\begin{array}{ll}{x^{2},} & {x \leq 1} \\ {-x^{2},} & {x>1}\end{array}\right. $$

Physical Science The temperature \(T\) (in degrees Fahrenheit) of food placed in a refrigerator is modeled by $$T=10\left(\frac{4 t^{2}+16 t+75}{t^{2}+4 t+10}\right)$$ where \(t\) is the time (in hours). What is the initial temperature of the food? Find the rates of change of \(T\) with respect to \(t\) when (a) \(t=1,\) (b) \(t=3,\) (c) \(t=5,\) and (d) \(t=10\).

Use the Constant Rule, the Constant Multiple Rule, and the Sum Rule to find \(h^{\prime}(1)\) given that \(f^{\prime}(1)=3\). $$ \text { (a) } h(x)=f(x)-2 \quad \text { (b) } h(x)=2 f(x) $$ $$ \text { (c) } h(x)=-f(x) \quad \text { (d) } h(x)=-1+2 f(x) $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x(3 x-9)^{3} $$

Describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\frac{1}{x-1} $$

Population Growth A population of bacteria is introduced into a culture. The number of bacteria \(P\) can be modeled by $$P=500\left(1+\frac{4 t}{50+t^{2}}\right)$$ where \(t\) is the time (in hours). Find the rate of change of the population when \(t=2\).

Revenue The revenue \(R\) (in millions of dollars per year) for Polo Ralph Lauren from 1999 through 2005 can be modeled by $$ \begin{aligned} R=0.59221 t^{4} &-18.0042 t^{3}+175.293 t^{2}-316.42 t \\\ &-116.5 \end{aligned} $$ where \(t\) is the year, with \(t=9\) corresponding to 1999 (a) Find the slopes of the graph for the years 2002 and 2004 . (b) Compare your results with those obtained in Exercise 11 in Section 2.1 (c) What are the units for the slope of the graph? Interpret the slope of the graph in the context of the problem.

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=x^{3}(x-4)^{2} $$

Describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\left\\{\begin{array}{ll}{x^{2}-3,} & {x \leq 0} \\ {3-x^{2},} & {x>0}\end{array}\right. $$

Quality Control The percent \(P\) of defective parts produced by a new employee \(t\) days after the employee starts work can be modeled by $$P=\frac{t+1750}{50(t+2)}$$ Find the rates of change of \(P\) when (a) \(t=1\) and (b) \(t=10\).

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=x \sqrt{2 x+3} $$

Identify a function that has the given characteristics. Then sketch the function. $$ f(0)=2 ; f^{\prime}(x)=-3,-\infty

MAKE A DECISION: NEGOTIATING A PRICE You decide to form a partnership with another business. Your business determines that the demand \(x\) for your product is inversely proportional to the square of the price for \(x \geq 5.\) (a) The price is \(\$ 1000\) and the demand is 16 units. Find the demand function. (b) Your partner determines that the product costs \(\$ 250\) per unit and the fixed cost is \(\$ 10,000 .\) Find the cost function. (c) Find the profit function and use a graphing utility to graph it. From the graph, what price would you negotiate with your partner for this product? Explain your reasoning.

Cost The variable cost for manufacturing an electrical component is \(\$ 7.75\) per unit, and the fixed cost is \(\$ 500 .\) Write the cost \(C\) as a function of \(x,\) the number of units produced. Show that the derivative of this cost function is a constant and is equal to the variable cost.

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