Chapter 9

Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=2 u^{2}+1\) and \(u=x^{2}+1\)

Find the third derivative of the given function. \(f(x)=\frac{2}{x^{2}}\)

VELOCITY OF BLOOD IN AN ARTERY The velocity (in centimeters/second) of blood \(r \mathrm{~cm}\) from the central axis of an artery is given by $$ v(r)=k\left(R^{2}-r^{2}\right) $$ where \(k\) is a constant and \(R\) is the radius of the artery (see the accompanying figure). Suppose \(k=1000\) and \(R=\) \(0.2 \mathrm{~cm}\). Find \(v(0.1)\) and \(v^{\prime}(0.1)\) and interpret your results.

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{x-1}{x^{2}+2 x-3}\)

Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=\sqrt{u}+\frac{1}{\sqrt{u}}\) and \(u=x^{3}-x\)

Find an equation of the tangent line to the graph of the function \(f(x)=\left(x^{3}+1\right)\left(3 x^{2}-4 x+2\right)\) at the point \((1,2)\).

SALES OF DIGITAL CAMERAS According to projections made in 2004 , the worldwide shipments of digital point-and-shoot cameras are expected to grow in accordance with the rule $$ N(t)=16.3 t^{0.8766} \quad(1 \leq t \leq 6) $$ where \(N(t)\) is measured in millions and \(t\) is measured in years, with \(t=1\) corresponding to 2001 . a. How many digital cameras were sold in \(2001(t=1)\) ? b. How fast were sales increasing in \(2001 ?\) c. What were the projected sales in \(2005 ?\) d. How fast were the sales projected to grow in 2005 ?

The distance \(s\) (in feet) covered by a motorcycle traveling in a straight line and starting from rest in \(t\) sec is given by the function $$ s(t)=-0.1 t^{3}+2 t^{2}+24 t $$ Calculate the motorcycle's average velocity over the time interval \([2,2+h]\) for \(h=1,0.1,0.01,0.001,0.0001\), and \(0.00001\) and use your results to guess at the motorcycle's instantaneous velocity at \(t=2\).

Find the values of \(x\) for which each function is continuous. \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x \leq 1 \\ 2 x-1 & \text { if } x>1\end{array}\right.\)

Find \(\frac{d y}{d u^{\prime}} \frac{d u}{d x^{\prime}}\) and \(\frac{d y}{d x}\). \(y=\frac{1}{u}\) and \(u=\sqrt{x}+1\)

Find an equation of the tangent line to the graph of the function \(f(x)=\frac{3 x}{x^{2}-2}\) at the point \((2,3)\).

ONLINE BUYERS As use of the Internet grows, so does the number of consumers who shop online. The number of online buyers, as a percent of net users, is expected to be $$ P(t)=53 t^{0.12} \quad(1 \leq t \leq 7) $$ where \(t\) is measured in years, with \(t=1\) corresponding to the beginning of 2002 . a. How many online buyers, as a percentage of net users, were there at the beginning of 2007 ? b. How fast was the number of online buyers, as a percentage of net users, changing at the beginning of 2007 ?

Find the values of \(x\) for which each function is continuous. \(f(x)=\left\\{\begin{array}{ll}-2 x+1 & \text { if } x<0 \\ x^{2}+1 & \text { if } x \geq 0\end{array}\right.\)

Find the indicated limit, if it exists. \(\lim _{b \rightarrow-3} \frac{b+1}{b+3}\)

Suppose \(F(x)=g(f(x))\) and \(f(2)=3, f^{\prime}(2)=-3\), \(g(3)=5\), and \(g^{\prime}(3)=4\). Find \(F^{\prime}(2)\)

Let \(f(x)=\left(x^{2}+1\right)(2-x)\). Find the point(s) on the graph of \(f\) where the tangent line is horizontal.

MARRIED HoUSEHOLDS WITH CHILDREN The percentage of families that were married households with children between 1970 and 2000 is approximately $$ P(t)=\frac{49.6}{t^{0.27}} \quad(1 \leq t \leq 4) $$ where \(t\) is measured in decades, with \(t=1\) corresponding to \(1970 .\) a. What percentage of families were married households with children in \(1970 ?\) In \(1980 ?\) In \(1990 ?\) In 2000 ? b. How fast was the percentage of families that were married households with children changing in \(1980 ?\) In \(1990 ?\)

Find the values of \(x\) for which each function is continuous. \(f(x)=|x+1|\)

Suppose \(h=f \circ g\). Find \(h^{\prime}(0)\) given that \(f(0)=6\), \(f^{\prime}(5)=-2, g(0)=5\), and \(g^{\prime}(0)=3\)

Let \(f(x)=\frac{x}{x^{2}+1} .\) Find the point(s) on the graph of \(f\) where the tangent line is horizontal.

EFFECT OF STOPPING ON AVERAGE SPEED According to data from a study, the average speed of your trip \(A\) (in \(\mathrm{mph}\) ) is related to the number of stops/mile you make on the trip \(x\) by the equation $$ A=\frac{26.5}{x^{0.45}} $$ Compute \(d A / d x\) for \(x=0.25\) and \(x=2\). How is the rate of change of the average speed of your trip affected by the number of stops/mile?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(f\) is continuous at \(x=a\) and \(g\) is differentiable at \(x=a\), then \(\lim _{x \rightarrow a} f(x) g(x)=f(a) g(a)\).

Find the values of \(x\) for which each function is continuous. \(f(x)=\frac{|x-1|}{x-1}\)

Suppose \(F(x)=f\left(x^{2}+1\right)\). Find \(F^{\prime}(1)\) if \(f^{\prime}(2)=3\).

Find the point(s) on the graph of the function \(f(x)=\) \(\left(x^{2}+6\right)(x-5)\) where the slope of the tangent line is equal to \(-2\).

Sketch the graph of the function \(f(x)=|x+1|\) and show that the function does not have a derivative at \(x=-1\).

In Exercises 57-60, determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{2 x}{x^{2}-1}\)

Find the indicated limit, if it exists. \(\lim _{x \rightarrow-2} \frac{x^{2}-x-6}{x^{2}+x-2}\)

Find the point \((s)\) on the graph of the function \(f(x)=\frac{x+1}{x-1}\) where the slope of the tangent line is equal to \(-\frac{1}{2}\).

DEMAND FuNCTIONS The demand function for the Luminar desk lamp is given by $$ p=f(x)=-0.1 x^{2}-0.4 x+35 $$ where \(x\) is the quantity demanded in thousands and \(p\) is the unit price in dollars. a. Find \(f^{\prime}(x)\). b. What is the rate of change of the unit price when the quantity demanded is 10,000 units \((x=10)\) ? What is the unit price at that level of demand?

Determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{1}{(x-1)(x-2)}\)

The concentration of a certain drug in a patient's bloodstream \(t\) hr after injection is given by $$ C(t)=\frac{0.2 t}{t^{2}+1} $$ a. Find the rate at which the concentration of the drug is changing with respect to time. b. How fast is the concentration changing \(\frac{1}{2} \mathrm{hr}, 1 \mathrm{hr}\), and 2 hr after the injection?

STOPPING DISTANCE OF A RACING CAR During a test by the editors of an auto magazine, the stopping distance \(s\) (in feet) of the MacPherson \(\mathrm{X}-2\) racing car conformed to the rule $$ s=f(t)=120 t-15 t^{2} \quad(t \geq 0) $$ where \(t\) was the time (in seconds) after the brakes were applied. a. Find an expression for the car's velocity \(v\) at any time \(t\). b. What was the car's velocity when the brakes were first applied? c. What was the car's stopping distance for that particular test? Hint: The stopping time is found by setting \(v=0\).

Let $$ f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \leq 1 \\ a x+b & \text { if } x>1 \end{array}\right. $$ Find the values of \(a\) and \(b\) so that \(f\) is continuous and has a derivative at \(x=1\). Sketch the graph of \(f\).

Determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{x^{2}-2 x}{x^{2}-3 x+2}\)

Find the indicated limit, if it exists. \(\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}\) Hint: Multiply by \(\frac{\sqrt{x}+1}{\sqrt{x}+1}\)

Suppose \(h=f \circ g .\) Show that \(h^{\prime}=\left(f^{\prime} \circ g\right) g^{\prime}\).

A city's main well was recently found to be contaminated with trichloroethylene, a cancer-causing chemical, as a result of an abandoned chemical dump leaching chemicals into the water. A proposal submitted to the city's council members indicates that the cost, measured in millions of dollars, of removing \(x \%\) of the toxic pollutant is given by $$ C(x)=\frac{0.5 x}{100-x} $$ Find \(C^{\prime}(80), C^{\prime}(90), C^{\prime}(95)\), and \(C^{\prime}(99)\). What does your result tell you about the cost of removing all of the pollutant?

INSTANT MESSAGING AccouNTS Mobile instant messaging (IM) is a small portion of total IM usage, but it is expected to grow sharply. The function $$ P(t)=0.257 t^{2}+0.57 t+3.9 \quad(0 \leq t \leq 4) $$ gives the projected mobile IM accounts as a percentage of total enterprise IM accounts from \(2006(t=0)\) through \(2010(t=4) .\) a. What percentage of total enterprise IM accounts are the mobile accounts expected to be in 2008 ? b. How fast is this percentage expected to change in 2008 ? Source: The Radical Group

Sketch the graph of the function \(f(x)=x^{2 / 3}\). Is the function continuous at \(x=0 ?\) Does \(f^{\prime}(0)\) exist? Why or why not?

Determine all values of \(x\) at which the function is discontinuous. \(f(x)=\frac{x^{2}-3 x+2}{x^{2}-2 x}\)

Find the indicated limit, if it exists. \(\lim _{x \rightarrow 4} \frac{x-4}{\sqrt{x}-2}\) Hint: See Exercise 59 .

In Exercises 61-64, find an equation of the tangent line to the graph of the function at the given point. \(f(x)=(1-x)\left(x^{2}-1\right)^{2} ;(2,-9)\)

Thomas Young has suggested the following rule for calculating the dosage of medicine for children 1 to 12 yr old. If \(a\) denotes the adult dosage (in milligrams) and if \(t\) is the child's age (in years), then the child's dosage is given by $$ D(t)=\frac{a t}{t+12} $$ Suppose the adult dosage of a substance is \(500 \mathrm{mg}\). Find an expression that gives the rate of change of a child's dosage with respect to the child's age. What is the rate of change of a child's dosage with respect to his or her age for a 6-yrold child? A 10 -yr-old child?

CHILD OBESITY The percentage of obese children, ages \(12-19\), in the United States has grown dramatically in recent years. The percentage of obese children from 1980 through the year 2000 is approximated by the function $$ P(t)=-0.0105 t^{2}+0.735 t+5 \quad(0 \leq t \leq 20) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1980 . a. What percentage of children were obese at the beginning of \(1980 ?\) At the beginning of \(1990 ?\) At the beginning of the year \(2000 ?\) b. How fast was the percentage of obese children changing at the beginning of \(1985 ?\) At the beginning of \(1990 ?\)

Prove that the derivative of the function \(f(x)=|x|\) for \(x \neq 0\) is given by $$ f^{\prime}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x>0 \\ -1 & \text { if } x<0 \end{array}\right. $$ Hint: Recall the definition of the absolute value of a number.

The graph of the "postage function" for 2008 , $$ f(x)=\left\\{\begin{array}{cl} 117 & \text { if } 0

Find an equation of the tangent line to the graph of the function at the given point. \(f(x)=\left(\frac{x+1}{x-1}\right)^{2} ;(3,4)\)

SPENDING ON MEDICARE Based on the current eligibility requirement, a study conducted in 2004 showed that federal spending on entitlement programs, particularly Medicare, would grow enormously in the future. The study predicted that spending on Medicare, as a percentage of the gross domestic product (GDP), will be $$ P(t)=0.27 t^{2}+1.4 t+2.2 \quad(0 \leq t \leq 5) $$ percent in year \(t\), where \(t\) is measured in decades, with \(t=0\) corresponding to 2000 . a. How fast will the spending on Medicare, as a percentage of the GDP, be growing in \(2010 ?\) In 2020 ? b. What will the predicted spending on Medicare be in \(2010 ?\) In \(2020 ?\)

Show that if a function \(f\) is differentiable at \(x=a\), then \(f\) must be continuous at that number. Hint: Write $$ f(x)-f(a)=\left[\frac{f(x)-f(a)}{x-a}\right](x-a) $$ Use the product rule for limits and the definition of the derivative to show that $$ \lim _{x \rightarrow a}[f(x)-f(a)]=0 $$