Chapter 7

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

find the limit $$ \lim _{x \rightarrow 3} f(x), \text { where } f(x)=\left\\{\begin{array}{ll} \frac{1}{3} x-2, & x \leq 3 \\ -2 x+5, & x>3 \end{array}\right. $$

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

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. $$ y=x^{3}+3 x^{2} $$

Describe the interval(s) on which the function is continuous. \(f(x)=x \sqrt{x+3}\)

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

find the limit $$ \lim _{x \rightarrow 1} f(s), \text { where } f(s)=\left\\{\begin{array}{ll} s, & s \leq 1 \\ 1-s, & s>1 \end{array}\right. $$

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

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. $$ y=\frac{1}{2} x^{2}+5 x $$

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

find the limit $$ \lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x} $$

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

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

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

Describe the interval(s) on which the function is continuous. \(f(x)=\frac{x+1}{\sqrt{x}}\)

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

find the limit $$ \lim _{\Delta x \rightarrow 0} \frac{4(x+\Delta x)-5-(4 x-5)}{\Delta x} $$

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

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 (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 \(g\) '. $$ \begin{aligned} &f(x)=x^{3} \\ &g(x)=x^{3}+3 \end{aligned} $$

Use a graphing utility to graph the function on the interval \([-4,4]\). Does the graph of the function appear to be continuous on this interval? Is the function in fact continuous on \([-4,4] ?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. \(f(x)=\frac{x^{2}+x}{x}\)

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

find the limit $$ \lim _{\Delta x \rightarrow 0} \frac{\sqrt{x+2+\Delta x}-\sqrt{x+2}}{\Delta x} $$

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

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) \(t=10\).

(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 \(g\) '. $$ \begin{aligned} &f(x)=x^{2} \\ &g(x)=3 x^{2} \end{aligned} $$

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

find the limit $$ \lim _{\Delta x \rightarrow 0} \frac{\sqrt{x+\Delta x}-\sqrt{x}}{\Delta x} $$

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

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 \(t=2\).

A deposit of $$\$ 7500$$ is made in an account that pays \(6 \%\) compounded quarterly. The amount \(A\) in the account after \(t\) years is \(A=7500(1.015)^{[4 t]}, \quad t \geq 0\) (a) Sketch the graph of \(A\). Is the graph continuous? Explain your reasoning. (b) What is the balance after 7 years?

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

find the limit $$ \lim _{\Delta t \rightarrow 0} \frac{(t+\Delta t)^{2}-5(t+\Delta t)-\left(t^{2}-5 t\right)}{\Delta t} $$

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

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) \(t=10\).

The cost \(C\) (in millions of dollars) of removing \(x\) percent of the pollutants emitted from the smokestack of a factory can be modeled by \(C=\frac{2 x}{100-x}\) (a) What is the implied domain of \(C ?\) Explain your reasoning. (b) Use a graphing utility to graph the cost function. Is the function continuous on its domain? Explain your reasoning. (c) Find the cost of removing \(75 \%\) of the pollutants from the smokestack.

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

find the limit $$ \lim _{\Delta r \rightarrow 0} \frac{(t+\Delta t)^{2}-4(t+\Delta t)+2-\left(t^{2}-4 t+2\right)}{\Delta t} $$

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

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.

A shipping company's charge for sending an overnight package from New York to Atlanta is \(\$ 12.80\) for the first pound and \(\$ 2.50\) for each additional pound or fraction thereof. Use the greatest integer function to create a model for the charge \(C\) for overnight delivery of a package weighing \(x\) pounds. Use a graphing utility to graph the function, and discuss its continuity.

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

Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 1^{-}} \frac{2}{x^{2}-1} $$

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

You are managing a store and have been adjusting the price of an item. You have found that you make a profit of \(\$ 50\) when 10 units are sold, \(\$ 60\) when 12 units are sold, and \(\$ 65\) when 14 units are sold. (a) Fit these data to the model \(P=a x^{2}+b x+c\). (b) Use a graphing utility to graph \(P\). (c) Find the point on the graph at which the marginal profit is zero. Interpret this point in the context of the problem.

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.

The United States Postal Service first class mail rates are $$\$ 0.41$$ for the first ounce and $$\$ 0.17$$ for each additional ounce or fraction thereof up to \(3.5\) ounces. A model for the cost \(C\) (in dollars) of a first class mailing that weighs \(3.5\) ounces or less is given below. (\mathrm{\\{} S o u r c e : ~ United States Postal Service) \(C(x)=\left\\{\begin{array}{ll}0.41, & 0 \leq x \leq 1 \\ 0.58, & 1

Identify a function \(f\) that has the given characteristics. Then sketch the function. $$ \begin{aligned} &f(-2)=f(4)=0 ; f^{\prime}(1)=0, f^{\prime}(x)<0 \\ &\text { for } x<1 ; f^{\prime}(x)>0 \text { for } x>1 \end{aligned} $$