Chapter 9

Find the indicated limit, if it exists. \(\lim _{x \rightarrow-2} \frac{4-x^{2}}{2 x^{2}+x^{3}}\)

find an equation of the tangent line to the graph of the function at the given point. \(f(x)=x \sqrt{2 x^{2}+7} ;(3,15)\)

FISHERIES The total groundfish population on Georges Bank in New England between 1989 and 1999 is approximated by the function $$ f(t)=5.303 t^{2}-53.977 t+253.8 \quad(0 \leq t \leq 10) $$ where \(f(t)\) is measured in thousands of metric tons and \(t\) in years, with \(t=0\) corresponding to the beginning of 1989 . a. What was the rate of change of the groundfish population at the beginning of \(1994 ?\) At the beginning of 1996 ? b. Fishing restrictions were imposed on Dec. 7,1994 . Were the conservation measures effective?

From experience, Emory Secretarial School knows that the average student taking Advanced Typing will progress according to the rule $$ N(t)=\frac{60 t+180}{t+6} \quad(t \geq 0) $$ where \(N(t)\) measures the number of words/minute the student can type after \(t\) wk in the course. a. Find an expression for \(N^{\prime}(t)\). b. Compute \(N^{\prime}(t)\) for \(t=1,3,4\), and 7 and interpret your results. c. Sketch the graph of the function \(N\). Does it confirm the results obtained in part (b)? d. What will be the average student's typing speed at the end of the 12 -wk course?

WORKER EFFICIENCY An efficiency study conducted for Elektra Electronics showed that the number of Space Commander walkie-talkies assembled by the average worker \(t\) hr after starting work at 8 a.m. is given by $$ N(t)=-t^{3}+6 t^{2}+15 t $$ a. Find the rate at which the average worker will be assembling walkie-talkies \(t\) hr after starting work. b. At what rate will the average worker be assembling walkie-talkies at 10 a.m.? At 11 a.m.? c. How many walkie-talkies will the average worker assemble between 10 a.m. and 11 a.m.?

TELEVISION VIEWING The number of viewers of a television series introduced several years ago is approximated by the function $$ N(t)=(60+2 t)^{2 / 3} \quad(1 \leq t \leq 26) $$ where \(N(t)\) (measured in millions) denotes the number of weekly viewers of the series in the \(t\) heek. Find the rate of increase of the weekly audience at the end of week 2 and at the end of week \(12 .\) How many viewers were there in week 2 ? In week 24 ?

The total worldwide box-office receipts for a long-running movie are approximated by the function $$ T(x)=\frac{120 x^{2}}{x^{2}+4} $$ where \(T(x)\) is measured in millions of dollars and \(x\) is the number of years since the movie's release. How fast are the total receipts changing \(1 \mathrm{yr}, 3 \mathrm{yr}\), and \(5 \mathrm{yr}\) after its release?

CONSUMER PRICE INDEX An economy's consumer price index (CPI) is described by the function $$ I(t)=-0.2 t^{3}+3 t^{2}+100 \quad(0 \leq t \leq 10) $$ where \(t=0\) corresponds to 1998 . a. At what rate was the CPI changing in 2003? In \(2005 ?\) In \(2008 ?\) b. What was the average rate of increase in the CPI over the period from 2003 to 2008 ?

According to a study conducted in 2003, the total number of U.S. jobs that are projected to leave the country by year \(t\), where \(t=0\) corresponds to the beginning of 2000, is $$ N(t)=0.0018425(t+5)^{2.5} \quad(0 \leq t \leq 15) $$ where \(N(t)\) is measured in millions. How fast was the number of U.S. jobs that were outsourced changing at the beginning of \(2005 ?\) How fast will the number of U.S. jobs that are outsourced be changing at the beginning of \(2010(t=10)\) ? Source: Forrester Research

A study on formaldehyde levels in 900 homes indicates that emissions of various chemicals can decrease over time. The formaldehyde level (parts per million) in an average home in the study is given by $$ f(t)=\frac{0.055 t+0.26}{t+2} \quad(0 \leq t \leq 12) $$ where \(t\) is the age of the house in years. How fast is the formaldehyde level of the average house dropping when it is new? At the beginning of its fourth year?

EfFECT OF ADVERTISING ON SALES The relationship between the amount of money \(x\) that Cannon Precision Instruments spends on advertising and the company's total sales \(S(x)\) is given by the function \(S(x)=-0.002 x^{3}+0.6 x^{2}+x+500 \quad(0 \leq x \leq 200)\) where \(x\) is measured in thousands of dollars. Find the rate of change of the sales with respect to the amount of money spent on advertising. Are Cannon's total sales increasing at a faster rate when the amount of money spent on advertising is (a) \(\$ 100,000\) or (b) \(\$ 150,000\) ?

WoRKING MoTHERS The percentage of mothers who work outside the home and have children younger than age \(6 \mathrm{yr}\) is approximated by the function $$ P(t)=33.55(t+5)^{0.205} \quad(0 \leq t \leq 21) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of \(1980 .\) Compute \(P^{\prime}(t) .\) At what rate was the percentage of these mothers changing at the beginning of \(2000 ?\) What was the percentage of these mothers at the beginning of \(2000 ?\)

A major corporation is building a 4325-acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Cove's population (in thousands) \(t\) yr from now will be given by $$ P(t)=\frac{25 t^{2}+125 t+200}{t^{2}+5 t+40} $$ a. Find the rate at which Glen Cove's population is changing with respect to time. b. What will be the population after 10 yr? At what rate will the population be increasing when \(t=10\) ?

SuPPLY FuNCTIONS The supply function for a certain make of satellite radio is given by $$ p=f(x)=0.0001 x^{5 / 4}+10 $$ where \(x\) is the quantity supplied and \(p\) is the unit price in dollars. a. Find \(f^{\prime}(x)\). b. What is the rate of change of the unit price if the quantity supplied is 10,000 satellite radios?

A dextrose solution is being administered to a patient intravenously. The 1-liter (L) bottle holding the solution is removed and replaced by another as soon as the contents drop to approximately \(5 \%\) of the initial (1-L) amount. The rate of discharge is constant, and it takes \(6 \mathrm{hr}\) to discharge \(95 \%\) of the contents of a full bottle. Draw a graph showing the amount of dextrose solution in a bottle in the IV system over a 24 -hr period, assuming that we started with a full bottle.

SELLING PRICE OF DVD RECORDERS The rise of digital music and the improvement to the DVD format are some of the reasons why the average selling price of standalone DVD recorders will drop in the coming years. The function $$ A(t)=\frac{699}{(t+1)^{0.94}} \quad(0 \leq t \leq 5) $$ gives the projected average selling price (in dollars) of stand-alone DVD recorders in year \(t\), where \(t=0\) corresponds to the beginning of 2002 . How fast was the average selling price of standalone DVD recorders falling at the beginning of \(2002 ?\) How fast was it falling at the beginning of \(2006 ?\)

The distance \(s\) (in feet) covered by a car \(t\) sec after starting from rest is given by $$ s=-t^{3}+8 t^{2}+20 t \quad(0 \leq t \leq 6) $$ Find a general expression for the car's acceleration at any time \(t(0 \leq t \leq 6)\). Show that the car is decelerating \(2 \frac{2}{3}\) sec after starting from rest.

PopULATION GRoWTH A study prepared for a Sunbelt town's chamber of commerce projected that the town's population in the next 3 yr will grow according to the rule $$ P(t)=50,000+30 t^{3 / 2}+20 t $$ where \(P(t)\) denotes the population \(t\) mo from now. How fast will the population be increasing \(9 \mathrm{mo}\) and \(16 \mathrm{mo}\) from now?

The base monthly salary of a salesman working on commission is \(\$ 22,000\). For each \(\$ 50,000\) of sales beyond \(\$ 100,000\), he is paid a \(\$ 1000\) commission. Sketch a graph showing his earnings as a function of the level of his sales \(x\). Determine the values of \(x\) for which the function \(\bar{f}\) is discontinuous.

SocIALLY RESPONSIBLE FuNDS Since its inception in 1971 , socially responsible investments, or SRIs, have yielded returns to investors on par with investments in general. The assets of socially responsible funds (in billions of dollars) from 1991 through 2001 is given by $$ f(t)=23.7(0.2 t+1)^{1.32} \quad(0 \leq t \leq 11) $$ where \(t=0\) corresponds to the beginning of 1991 . a. Find the rate at which the assets of SRIs were changing at the beginning of 2000 . b. What were the assets of SRIs at the beginning of 2000 ?

CRIME RATES The number of major crimes committed in Bronxville between 2000 and 2007 is approximated by the function $$ N(t)=-0.1 t^{3}+1.5 t^{2}+100 \quad(0 \leq t \leq 7) $$ where \(N(t)\) denotes the number of crimes committed in year \(t\), with \(t=0\) corresponding to the beginning of 2000 . Enraged by the dramatic increase in the crime rate, Bronxville's citizens, with the help of the local police, organized "Neighborhood Crime Watch" groups in early 2004 to combat this menace. a. Verify that the crime rate was increasing from the beginning of 2000 to the beginning of 2007 . Hint: Compute \(N^{\prime}(0), N^{\prime}(1), \ldots, N^{\prime}(7)\). b. Show that the Neighborhood Crime Watch program was working by computing \(N^{\prime \prime}(4), N^{\prime \prime}(5), N^{\prime \prime}(6)\), and \(N^{\prime \prime}(7) .\)

AVERAGE SPEED OF A VEHICLE ON A HIGHWAY The average speed of a vehicle on a stretch of Route 134 between 6 a.m. and 10 a.m. on a typical weekday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+50 \quad(0 \leq t \leq 4) $$ where \(f(t)\) is measured in mph and \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. a. Compute \(f^{\prime}(t)\). b. What is the average speed of a vehicle on that stretch of Route 134 at 6 a.m.? At 7 a.m.? At 8 a.m.? c. How fast is the average speed of a vehicle on that stretch of Route 134 changing at \(6: 30\) a.m.? At 7 a.m.? At 8 a.m.?

The fee charged per car in a downtown parking lot is \(\$ 2.00\) for the first half hour and \(\$ 1.00\) for each additional half hour or part thereof, subject to a maximum of \(\$ 10.00\). Derive a function \(f\) relating the parking fee to the length of time a car is left in the lot. Sketch the graph of \(f\) and determine the values of \(x\) for which the function \(f\) is discontinuous.

In Exercises 69-72, complete the table by computing \(f(x)\) at the given values of \(x\). Use the results to guess at the indicated limits, if they exist. $$ f(x)=\frac{1}{x^{2}+1} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) $$

AGING PoPULATION The population of Americans age \(55 \mathrm{yr}\) and older as a percentage of the total population is approximated by the function $$ f(t)=10.72(0.9 t+10)^{0.3} \quad(0 \leq t \leq 20) $$ where \(t\) is measured in years, with \(t=0\) corresponding to the year 2000 . At what rate was the percentage of Americans age 55 and over changing at the beginning of \(2000 ?\) At what rate will the percentage of Americans age \(55 \mathrm{yr}\) and older be changing in \(2010 ?\) What will be the percentage of the population of Americans age \(55 \mathrm{yr}\) and older in 2010 ?

GDP OF A DEVELOPING CoUNTRY A developing country's gross domestic product (GDP) from 2000 to 2008 is approximated by the function $$ G(t)=-0.2 t^{3}+2.4 t^{2}+60 \quad(0 \leq t \leq 8) $$ where \(G(t)\) is measured in billions of dollars, with \(t=0\) corresponding to the beginning of 2000 . a. Compute \(G^{\prime}(0), G^{\prime}(1), \ldots, G^{\prime}(8)\). b. Compute \(G^{\prime \prime}(0), G^{\prime \prime}(1), \ldots, G^{\prime \prime}(8)\). c. Using the results obtained in parts (a) and (b), show that after a spectacular growth rate in the early years, the growth of the GDP cooled off.

CURBING POPULATION GROWTH Five years ago, the government of a Pacific Island state launched an extensive propaganda campaign toward curbing the country's population growth. According to the Census Department, the population (measured in thousands of people) for the following 4 yr was $$ P(t)=-\frac{1}{3} t^{3}+64 t+3000 $$ where \(t\) is measured in years and \(t=0\) corresponds to the start of the campaign. Find the rate of change of the population at the end of years \(1,2,3\), and 4 . Was the plan working?

The function that gives the cost of a certain commodity is defined by $$ C(x)=\left\\{\begin{array}{ll} 5 x & \text { if } 0

Complete the table by computing \(f(x)\) at the given values of \(x\). Use the results to guess at the indicated limits, if they exist. $$ f(x)=\frac{2 x}{x+1} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) $$

CONCENTRATION OF CARBON MoNOXIDE (CO) IN THE AIR According to a joint study conducted by Oxnard's Environmental Management Department and a state government agency, the concentration of \(\mathrm{CO}\) in the air due to automobile exhaust \(t\) yr from now is given by $$ C(t)=0.01\left(0.2 t^{2}+4 t+64\right)^{2 / 3} $$ parts per million. a. Find the rate at which the level of \(\mathrm{CO}\) is changing with respect to time. b. Find the rate at which the level of \(\mathrm{CO}\) will be changing 5 yr from now.

The number of persons aged \(18-64\) receiving disability benefits through Social Security, the Supplemental Security income, or both, from 1990 through 2000 is approximated by the function \(N(t)=0.00037 t^{3}-0.0242 t^{2}+0.52 t+5.3 \quad(0 \leq t \leq 10)\) where \(f(t)\) is measured in units of a million and \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1990 . Compute \(N(8), N^{\prime}(8)\), and \(N^{\prime \prime}(8)\) and interpret your results. Source: Social Security Administration

CONSERVATION OF SPECIES A certain species of turtle faces extinction because dealers collect truckloads of turtle eggs to be sold as aphrodisiacs. After severe conservation measures are implemented, it is hoped that the turtle population will grow according to the rule $$ N(t)=2 t^{3}+3 t^{2}-4 t+1000 \quad(0 \leq t \leq 10) $$ where \(N(t)\) denotes the population at the end of year \(t\). Find the rate of growth of the turtle population when \(t=2\) and \(t=8\). What will be the population 10 yr after the conservation measures are implemented?

According to Weiss's law of excitation of tissue, the strength \(S\) of an electric current is related to the time \(t\) the current takes to excite tissue by the formula $$ S(t)=\frac{a}{t}+b \quad(t>0) $$ where \(a\) and \(b\) are positive constants. a. Evaluate \(\lim _{t \rightarrow 0^{+}} S(t)\) and interpret your result. b. Evaluate \(\lim _{t \rightarrow \infty} S(t)\) and interpret your result. (Note: The limit in part (b) is called the threshold strength of the current. Why?)

Complete the table by computing \(f(x)\) at the given values of \(x\). Use the results to guess at the indicated limits, if they exist. $$ f(x)=3 x^{3}-x^{2}+10 ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) $$

CoNTINUING EDUCATION ENROLLMENT The registrar of Kellogg University estimates that the total student enrollment in the Continuing Education division will be given by $$ N(t)=-\frac{20,000}{\sqrt{1+0.2 t}}+21,000 $$ where \(N(t)\) denotes the number of students enrolled in the division \(t\) yr from now. Find an expression for \(N^{\prime}(t)\). How fast is the student enrollment increasing currently? How fast will it be increasing 5 yr from now?

The number of Americans aged 45 to 54 is approximately $$ \begin{aligned} N(t)=&-0.00233 t^{4}+0.00633 t^{3}-0.05417 t^{2} \\ &+1.3467 t+25 \end{aligned} $$ million people in year \(t\), with \(t=0\) corresponding to the beginning of 1990 . Compute \(N^{\prime}(10)\) and \(N^{\prime \prime}(10)\) and interpret your results. Source U.S. Census Bureau

FuGHT OF A RocKET The altitude (in feet) of a rocket \(t\) sec into flight is given by $$ s=f(t)=-2 t^{3}+114 t^{2}+480 t+1 \quad(t \geq 0) $$ a. Find an expression \(v\) for the rocket's velocity at any time \(t\). b. Compute the rocket's velocity when \(t=0,20,40\), and 60\. Interpret your results. c. Using the results from the solution to part (b), find the maximum altitude attained by the rocket. Hint: At its highest point, the velocity of the rocket is zero.

Suppose a fish swimming a distance of \(L \mathrm{ft}\) at a speed of \(v \mathrm{ft} / \mathrm{sec}\) relative to the water and against a current flowing at the rate of \(u \mathrm{ft} / \mathrm{sec}(u

Complete the table by computing \(f(x)\) at the given values of \(x\). Use the results to guess at the indicated limits, if they exist. $$ f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) $$

AIR PoLLuTION According to the South Coast Air Quality Management District, the level of nitrogen dioxide, a brown gas that impairs breathing, present in the atmosphere on a certain May day in downtown Los Angeles is approximated by $$ A(t)=0.03 t^{3}(t-7)^{4}+60.2 \quad(0 \leq t \leq 7) $$ where \(A(t)\) is measured in pollutant standard index and \(t\) is measured in hours, with \(t=0\) corresponding to 7 a.m.

The body mass index (BMI) measures body weight in relation to height. A BMI of 25 to \(29.9\) is considered overweight, a BMI of 30 or more is considered obese, and a BMI of 40 or more is morbidly obese. The percent of the U.S. population that is obese is approximated by the function \(P(t)=0.0004 t^{3}+0.0036 t^{2}+0.8 t+12 \quad(0 \leq t \leq 13)\) where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1991 . Show that the rate of the rate of change of the percent of the U.S. population that is deemed obese was positive from 1991 to 2004 . What does this mean?

OBESITY IN AMERICA The body mass index (BMI) measures body weight in relation to height. A BMI of 25 to \(29.9\) is considered overweight, a BMI of 30 or more is considered obese, and a BMI of 40 or more is morbidly obese. The percentage of the U.S. population that is obese is approximated by the function \(P(t)=0.0004 t^{3}+0.0036 t^{2}+0.8 t+12 \quad(0 \leq t \leq 13)\) where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1991 . a. What percentage of the U.S. population was deemed obese at the beginning of \(1991 ?\) At the beginning of \(2004 ?\) b. How fast was the percentage of the U.S. population that is deemed obese changing at the beginning of \(1991 ?\) At the beginning of 2004 ? (Note: A formula for calculating the BMI of a person is given in Exercise 29, page \(834 .\) )

Let $$ f(x)=\left\\{\begin{array}{ll} x+2 & \text { if } x \leq 1 \\ k x^{2} & \text { if } x>1 \end{array}\right. $$ Find the value of \(k\) that will make \(f\) continuous on \((-\infty, \infty)\).

In Exercises 73-80, find the indicated limits, if they exist. \(\lim _{x \rightarrow \infty} \frac{3 x+2}{x-5}\)

EFFECT OF LUXURY TAX ON CONSUMPTION Government economists of a developing country determined that the purchase of imported perfume is related to a proposed "luxury tax" by the formula \(N(x)=\sqrt{10,000-40 x-0.02 x^{2}} \quad(0 \leq x \leq 200)\) where \(N(x)\) measures the percentage of normal consumption of perfume when a "luxury tax" of \(x \%\) is imposed on it. Find the rate of change of \(N(x)\) for taxes of \(10 \%, 100 \%\), and \(150 \%\).

AIR PURIFICATION During testing of a certain brand of air purifier, the amount of smoke remaining \(t\) min after the start of the test was $$ \begin{aligned} A(t)=&-0.00006 t^{5}+0.00468 t^{4}-0.1316 t^{3} \\ &+1.915 t^{2}-17.63 t+100 \end{aligned} $$ percent of the original amount. Compute \(A^{\prime}(10)\) and \(A^{\prime \prime}(10)\) and interpret your results. Source: Consumer Reports

HEALTH-CARE SPENDING Despite efforts at cost containment, the cost of the Medicare program is increasing. Two major reasons for this increase are an aging population and extensive use by physicians of new technologies. Based on data from the Health Care Financing Administration and the U.S. Census Bureau, health-care spending through the year 2000 may be approximated by the function $$ \begin{array}{l} S(t)=0.02836 t^{3}-0.05167 t^{2}+9.60881 t+41.9 \\ \quad(0 \leq t \leq 35) \end{array} $$ where \(S(t)\) is the spending in billions of dollars and \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1965 . a. Find an expression for the rate of change of health-care spending at any time \(t\). b. How fast was health-care spending changing at the beginning of 1980 ? At the beginning of 2000 ? c. What was the amount of health-care spending at the beginning of 1980 ? At the beginning of 2000 ?

Let $$ f(x)=\left\\{\begin{array}{ll} \frac{x^{2}-4}{x+2} & \text { if } x \neq-2 \\ k & \text { if } x=-2 \end{array}\right. $$ For what value of \(k\) will \(f\) be continuous on \((-\infty, \infty)\) ?

PULSE RATE OF AN ATHLETE The pulse rate (the number of heartbeats/minute) of a long-distance runner \(t\) sec after leaving the starting line is given by $$ P(t)=\frac{300 \sqrt{\frac{1}{2} t^{2}+2 t+25}}{t+25} \quad(t \geq 0) $$ Compute \(P^{\prime}(t)\). How fast is the athlete's pulse rate increasing \(10 \mathrm{sec}, 60 \mathrm{sec}\), and 2 min into the run? What is her pulse rate 2 min into the run?

AGING PoPULATION The population (in millions) of developed countries from 2005 through 2034 is projected to be $$ f(t)=3.567 t+175.2 \quad(5 \leq t \leq 35) $$ where \(t\) is measured in years. On the other hand, the population of underdeveloped/emerging countries over the same period is projected to be $$ g(t)=0.46 t^{2}+0.16 t+287.8 \quad(5 \leq t \leq 35) $$ a. What does the function \(D=g+f\) represent? b. Find \(D^{\prime}\) and \(D^{\prime}(10)\) and interpret your results.