Chapter 11

Suppose in a certain country the life expectancy at birth of a female is changing at the rate of $$ g^{\prime}(t)=\frac{5.45218}{(1+1.09 t)^{09}} $$ years/year. Here, \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1900 . Find an expression \(g(t)\) giving the life expectancy at birth (in years) of a female in that country if the life expectancy at the beginning of 1900 is \(50.02 \mathrm{yr}\). What is the life expectancy at birth of a female born in 2000 in that country?

Find the function \(f\) given that the slope of the tangent line to the graph of \(f\) at any point \((x, f(x))\) is \(f^{\prime}(x)\) and that the graph of \(f\) passes through the given point. $$f^{\prime}(x)=\frac{2}{x}+1 ;(1,2)$$

Using data collected at Kaiser Hospital, pediatricians estimate that the average height of male children changes at the rate of $$ h^{\prime}(t)=\frac{52.8706 e^{-0.327 t}}{\left(1+2.449 e^{-0.3277}\right)^{2}} $$ inches/year, where the child's height \(h(t)\) is measured in inches and \(t\), the child's age, is measured in years, with \(t=0\) corresponding to the age at birth. Find an expression \(h(t)\) for the average height of a boy at age \(t\) if the height at birth of an average child is \(19.4 \mathrm{in}\). What is the height of an average 8 -yr-old boy?

The average student enrolled in the 20-wk Court Reporting I course at the American Institute of Court Reporting progresses according to the rule $$ N^{\prime}(t)-6 e^{-0.05 t} \quad(0 \leq t \leq 20) $$ where \(N^{\prime}(t)\) measures the rate of change in the number of words/minute of dictation the student takes in machine shorthand after \(t\) wk in the course. Assuming that the average student enrolled in this course begins with a dictation speed of 60 words/minute, find an expression \(N(t)\) that gives the dictation speed of the student after \(t\) wk in the course.

Verify by direct computation that $$ \int_{1}^{3} x^{2} d x=-\int_{3}^{1} x^{2} d x $$

Suppose a patient is given a continuous intravenous infusion of glucose at a constant rate of \(r \mathrm{mg} / \mathrm{min}\). Then, the rate at which the amount of glucose in the bloodstream is changing at time \(t\) due to this infusion is given by $$ A^{\prime}(t)=r e^{-a t} $$ \(\mathrm{mg} / \mathrm{min}\), where \(a\) is a positive constant associated with the rate at which excess glucose is eliminated from the bloodstream and is dependent on the patient's metabolism rate. Derive an expression for the amount of glucose in the bloodstream at time \(t\).

The velocity of a car (in feet/second) \(t\) sec after starting from rest is given by the function $$ f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30) $$ Find the car's position, \(s(t)\), at any time \(t\). Assume \(s(0)=0\).

A drug is carried into an organ of volume \(V \mathrm{~cm}^{3}\) by a liquid that enters the organ at the rate of \(a \mathrm{~cm}^{3} / \mathrm{sec}\) and leaves it at the rate of \(b \mathrm{~cm}^{3} / \mathrm{sec} .\) The concentration of the drug in the liquid entering the organ is \(c \mathrm{~g} / \mathrm{cm}^{3}\). If the concentration of the drug in the organ at time \(t\) is increasing at the rate of $$ x^{\prime}(t)=\frac{1}{V}\left(a c-b x_{0}\right) e^{-h d V} $$ \(\mathrm{g} / \mathrm{cm}^{3} / \mathrm{sec}\), and the concentration of the drug in the organ initially is \(x_{0} \mathrm{~g} / \mathrm{cm}^{3}\), show that the concentration of the drug in the organ at time \(t\) is given by $$ x(t)=\frac{a c}{b}+\left(x_{0}-\frac{a c}{b}\right) e^{-b v v} $$

The velocity (in feet/second) of a maglev is $$ v(t)=0.2 t+3 \quad(0 \leq t \leq 120) $$ At \(t=0\), it is at the station. Find the function giving the position of the maglev at time \(t\), assuming that the motion takes place along a straight stretch of track.

The velocity of a car (in feet/second) \(t\) sec after starting from rest is given by the function $$ f(t)=2 \sqrt{t} \quad(0 \leq t \leq 30) $$ Find the car's position, \(s(t)\), at any time \(t\). Assume \(s(0)=0\).

Lorimar Watch Company manufactures travel clocks. The daily marginal cost function associated with producing these clocks is $$ C^{\prime}(x)=0.000009 x^{2}-0.009 x+8 $$ where \(C^{\prime}(x)\) is measured in dollars/unit and \(x\) denotes the number of units produced. Management has determined that the daily fixed cost incurred in producing these clocks is \(\$ 120 .\) Find the total cost incurred by Lorimar in producing the first 500 travel clocks/day.

Verify by direct computation that $$ \int_{0}^{1}\left(1+x-e^{x}\right) d x=\int_{0}^{1} d x+\int_{0}^{1} x d x-\int_{0}^{1} e^{x} d x $$ What properties of the definite integral are demonstrated in this exercise?

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, then \(\int f(a x+b) d x=a \int f(x) d x\).

The management of Lorimar Watch Company has determined that the daily marginal revenue function associated with producing and selling their travel clocks is given by $$ R^{\prime}(x)=-0.009 x+12 $$ where \(x\) denotes the number of units produced and sold and \(R^{\prime}(x)\) is measured in dollars/unit. a. Determine the revenue function \(R(x)\) associated with producing and selling these clocks. b. What is the demand equation that relates the wholesale unit price with the quantity of travel clocks demanded?

Verify by direct computation that $$ \int_{0}^{3}\left(1+x^{3}\right) d x=\int_{0}^{1}\left(1+x^{3}\right) d x+\int_{1}^{3}\left(1+x^{3}\right) d x $$ What property of the definite integral is demonstrated here?

Cannon Precision Instruments makes an automatic electronic flash with Thyrister circuitry. The estimated marginal profit associated with producing and selling these electronic flashes is $$ P^{\prime}(x)=-0.004 x+20 $$ dollars/unit/month when the production level is \(x\) units per month. Cannon's fixed cost for producing and selling these electronic flashes is \(\$ 16,000 /\) month. At what level of production does Cannon realize a maximum profit? What is the maximum monthly profit?

Verify by direct computation that \(\int_{0}^{3}\left(1+x^{3}\right) d x\) \(\quad=\int_{0}^{1}\left(1+x^{3}\right) d x+\int_{1}^{2}\left(1+x^{3}\right) d x+\int_{2}^{3}\left(1+x^{3}\right) d x\) hence showing that Property 5 may be extended.

Carlota Music Company estimates that the marginal cost of manufacturing its Professional Series guitars is $$ C^{\prime}(x)=0.002 x+100 $$ dollars/month when the level of production is \(x\) guitars/ month. The fixed costs incurred by Carlota are \(\$ 4000\) / month. Find the total monthly cost incurred by Carlota in manufacturing \(x\) guitars/month.

Evaluate \(\int_{3}^{3}(1+\sqrt{x}) e^{-x} d x\)

The national health expenditures are projected to grow at the rate of $$ r(t)=0.0058 t+0.159 \quad(0 \leq t \leq 13) $$ trillion dollars/year from 2002 through \(2015 .\) Here, \(t=0\) corresponds to 2002. The expenditure in 2002 was $$\$ 1.60$$ trillion. a. Find a function \(f\) giving the projected national health expenditures in year \(t\). b. What does your model project the national health expenditure to be in 2015 ?

Evaluate \(\int_{3}^{0} f(x) d x\), given that \(\int_{0}^{3} f(x) d x=4\).

As part of a quality-control program, the chess sets manufactured by Jones Brothers are subjected to a final inspection before packing. The rate of increase in the number of sets checked per hour by an inspector \(t\) hr into the \(8 \mathrm{a} . \mathrm{m}\). to 12 noon morning shift is approximately $$ N^{\prime}(t)=-3 t^{2}+12 t+45 \quad(0 \leq t \leq 4) $$ a. Find an expression \(N(t)\) that approximates the number of sets inspected at the end of \(t\) hours. Hint: \(N(0)=0\). b. How many sets does the average inspector check during a morning shift?

Given that \(\int_{-1}^{2} f(x) d x=-2\) and \(\int_{-1}^{2} g(x) d x=3\), evaluate a. \(\int_{-1}^{2}[2 f(x)+g(x)] d x\) b. \(\int_{-1}^{2}[g(x)-f(x)] d x\) c. \(\int_{-1}^{2}[2 f(x)-3 g(x)] d x\)

Based on data obtained by polling automobile buyers, the number of subscribers of satellite radios is expected to grow at the rate of $$ r(t)=-0.375 t^{2}+2.1 t+2.45 \quad(0 \leq t \leq 5) $$ million subscribers/year between \(2003(t=0)\) and 2008 \((t=5)\). The number of satellite radio subscribers at the beginning of 2003 was \(1.5\) million. a. Find an expression giving the number of satellite radio subscribers in year \(t(0 \leq t \leq 5)\). b. Based on this model, what was the number of satellite radio subscribers in 2008 ?

Given that \(\int_{-1}^{2} f(x) d x=2\) and \(\int_{0}^{2} f(x) d x=3\), evaluate a. \(\int_{-1}^{0} f(x) d x\) b. \(\int_{0}^{2} f(x) d x-\int_{-1}^{0} f(x) d x\)

The rate at which the risk of Down syndrome is changing is approximated by the function \(r(x)=0.004641 x^{2}-0.3012 x+4.9 \quad(20 \leq x \leq 45)\) where \(r(x)\) is measured in percentage of all births/year and \(x\) is the maternal age at delivery. a. Find a function \(f\) giving the risk as a percentage of all births when the maternal age at delivery is \(x\) years, given that the risk of down syndrome at age 30 is \(0.14 \%\) of all births. b. Based on this model, what is the risk of Down syndrome when the maternal age at delivery is 40 years? 45 years?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. $$\int_{2}^{2} \frac{e^{x}}{\sqrt{1+x}} d x=0$$

The total number of acres of genetically modified crops grown worldwide from 1997 through 2003 was changing at the rate of $$ R(t)=2.718 t^{2}-19.86 t+50.18 \quad(0 \leq t \leq 6) $$ million acres/year. The total number of acres of such crops grown in 1997 ( \(t=0\) ) was \(27.2\) million acres. How many acres of genetically modified crops were grown worldwide in \(2003 ?\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \(\int_{0}^{1} x \sqrt{x+1} d x=\sqrt{x+1} \int_{0}^{1} x d x=\left.\frac{1}{2} x^{2} \sqrt{x+1}\right|_{0} ^{1}=\frac{\sqrt{2}}{2}\)78. If \(f^{\prime}\) is continuous on \([0,2]\), then \(\int_{0}^{2} f^{\prime}(x) d x=f(2)-f(0)\).

One method of weight loss gaining in popularity is stomach-reducing surgery. It is generally reserved for people at least \(100 \mathrm{lb}\) overweight because the procedure carries a serious risk of death or complications. According to the American Society of Bariatric Surgery, the number of morbidly obese patients undergoing the procedure was increasing at the rate of $$ R(t)=9.399 t^{2}-13.4 t+14.07 \quad(0 \leq t \leq 3) $$ thousands/year, with \(t=0\) corresponding to 2000 . The number of gastric bypass surgeries performed in 2000 was \(36.7\) thousand. a, Find an expression giving the number of gastric bypass surgeries performed in year \(t(0 \leq t \leq 3)\). b. Use the result of part (a) to find the number of gastric bypass surgeries performed in 2003 .

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \(\int_{0}^{1} x \sqrt{x+1} d x=\sqrt{x+1} \int_{0}^{1} x d x=\left.\frac{1}{2} x^{2} \sqrt{x+1}\right|_{0} ^{1}=\frac{\sqrt{2}}{2}\)78. If \(f^{\prime}\) is continuous on \([0,2]\), then \(\int_{0}^{2} f^{\prime}(x) d x=f(2)-f(0)\).

According to a study conducted in 2004, the share of online advertisement, worldwide, as a percentage of the total ad market, is expected to grow at the rate of $$ R(t)=-0.033 t^{2}+0.3428 t+0.07 \quad(0 \leq t \leq 6) $$ percent/year at time \(t\) (in years), with \(t=0\) corresponding to the beginning of 2000 . The online ad market at the beginning of 2000 was \(2.9 \%\) of the total ad market. a. What is the projected online ad market share at any time \(t\) ? b. What was the projected online ad market share at the beginning of 2005 ?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f\) and \(g\) are continuous on \([a, b]\) and \(k\) is a constant, then $$ \int_{a}^{b}[k f(x)+g(x)] d x=k \int_{a}^{b} f(x) d x+\int_{a}^{b} g(x) d x $$

The average out-of-pocket costs for beneficiaries in traditional Medicare (including premiums, cost sharing, and prescription drugs not covered by Medicare) is projected to grow at the rate of $$ C^{\prime}(t)=12.288 t^{2}-150.5594 t+695.23 $$ dollars/year, where \(t\) is measured in 5 -yr intervals, with \(t=0\) corresponding to \(2000 .\) The out-of-pocket costs for beneficiaries in 2000 were $$\$ 3142$$. a. Find an expression giving the average out-of-pocket costs for beneficiaries in year \(t\). b. What is the projected average out-of-pocket costs for beneficiaries in 2010 ?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(f\) is continuous on \([a, b]\) and \(a

A ballast is dropped from a stationary hot-air balloon that is hovering at an altitude of \(400 \mathrm{ft}\). Its velocity after \(t\) sec is \(-32 t \mathrm{ft} / \mathrm{sec}\). a. Find the height \(h(t)\) of the ballast from the ground at time \(t\) Hint: \(h^{\prime}(t)=-32 t\) and \(h(0)=400\). b. When will the ballast strike the ground? c. Find the velocity of the ballast when it hits the ground.

A study conducted by TeleCable estimates that the number of cable TV subscribers will grow at the rate of $$ 100+210 t^{3 / 4} $$ new subscribers/month, \(t\) mo from the start date of the service. If 5000 subscribers signed up for the service before the starting date, how many subscribers will there be \(16 \mathrm{mo}\) from that date?

The rate of change of the level of ozone, an invisible gas that is an irritant and impairs breathing, present in the atmosphere on a certain May day in the city of Riverside is given by $$ R(t)=3.2922 t^{2}-0.366 t^{3} \quad(0

The velocity, in feet/second, of a rocket \(t\) sec into vertical flight is given by $$ v(t)=-3 t^{2}+192 t+120 $$ Find an expression \(h(t)\) that gives the rocket's altitude, in feet, \(t\) sec after liftoff. What is the altitude of the rocket 30 sec after liftoff?

The development of AstroWorld ("The Amusement Park of the Future") on the outskirts of a city will increase the city's population at the rate of $$ 4500 \sqrt{t}+1000 $$ people/year, \(t\) yr from the start of construction. The population before construction is \(30,000 .\) Determine the projected population 9 yr after construction of the park has begun.

The sales of organic milk from 1999 through 2004 grew at the rate of approximately $$ \begin{array}{r} R(t)=3 t^{3}-17.9445 t^{2}+28.7222 t+26.632 \\ (0 \leq t \leq 5) \end{array} $$ million dollars/year, where \(t\) is measured in years, with \(t=0\) corresponding to \(1999 .\) Sales of organic milk in 1999 totaled $$\$ 108$$ million. a. Find an expression giving the total sales of organic milk by year \(t(0 \leq t \leq 5)\). b. According to this model, what were the total sales of organic milk in \(2004 ?\)

Empirical data suggest that the surface area of a \(180-\mathrm{cm}\) -tall human body changes at the rate of $$ S^{\prime}(W)=0.131773 W^{-0.575} $$ square meters/kilogram, where \(W\) is the weight of the body in kilograms. If the surface area of a 180 -cm-tall human body weighing \(70 \mathrm{~kg}\) is \(1.886277 \mathrm{~m}^{2}\), what is the surface area of a human body of the same height weighing \(75 \mathrm{~kg}\) ?

The number of Medicarecertified home-health-care agencies ( \(70 \%\) are freestanding, and \(30 \%\) are owned by a hospital or other large facility) has been declining at the rate of $$ 0.186 e^{-0.02 t} \quad(0 \leq t \leq 14) $$ thousand agencies/year between \(1988(t=0)\) and 2002 \((t=14)\). The number of such agencies stood at \(9.3\) thousand units in 1988 . a. Find an expression giving the number of health-care agencies in year \(t\). b. What was the number of health-care agencies in \(2002 ?\) c. If this model held true through 2005 , how many care agencies were there in 2005 ?

According to the Jenss model for predicting the height of preschool children, the rate of growth of a typical preschool child is $$ R(t)=25.8931 e^{-0.993 t}+6.39 \quad\left(\frac{1}{4} \leq t \leq 6\right) $$ centimeters/year, where \(t\) is measured in years. The height of a typical 3 -mo-old preschool child is \(60.2952 \mathrm{~cm}\). a. Find a model for predicting the height of a typical preschool child at age \(t\). b. Use the result of part (a) to estimate the height of a typical 1-yr-old child.

A car traveling along a straight road at \(66 \mathrm{ft} / \mathrm{sec}\) accelerated to a speed of \(88 \mathrm{ft} / \mathrm{sec}\) over \(\mathrm{a}\) distance of \(440 \mathrm{ft}\). What was the acceleration of the car, assuming it was constant?

What constant deceleration would a car moving along a straight road have to be subjected to if it were brought to rest from a speed of \(88 \mathrm{ft} / \mathrm{sec}\) in \(9 \mathrm{sec}\) ? What would be the stopping distance?

A pilot lands a fighter aircraft on an aircraft carrier. At the moment of touchdown, the speed of the aircraft is \(160 \mathrm{mph}\). If the aircraft is brought to a complete stop in 1 sec and the deceleration is assumed to be constant, find the number of \(g\) 's the pilot is subjected to during landing \(\left(1 \mathrm{~g}=32 \mathrm{ft} / \mathrm{sec}^{2}\right) .\)

After rounding the final turn in the bell lap, two runners emerged ahead of the pack. When runner A is \(200 \mathrm{ft}\) from the finish line, his speed is \(22 \mathrm{ft} / \mathrm{sec}\), a speed that he maintains until he crosses the line. At that instant of time, runner \(\mathrm{B}\), who is \(20 \mathrm{ft}\) behind runner A and running at a speed of \(20 \mathrm{ft} / \mathrm{sec}\), begins to sprint. Assuming that runner B sprints with a constant acceleration, what minimum acceleration will enable him to cross the finish line ahead of runner A?

A tank has a constant cross-sectional area of \(50 \mathrm{ft}^{2}\) and an orifice of constant cross-sectional area of \(\frac{1}{2} \mathrm{ft}^{2}\) located at the bottom of the tank (see the accompanying figure). If the tank is filled with water to a height of \(h \mathrm{ft}\) and allowed to drain, then the height of the water decreases at a rate that is described by the equation $$ \frac{d h}{d t}=-\frac{1}{25}\left(\sqrt{20}-\frac{t}{50}\right) \quad(0 \leq t \leq 50 \sqrt{20}) $$ Find an expression for the height of the water at any time \(t\) if its height initially is \(20 \mathrm{ft}\)

During a thunderstorm, rain was falling at the rate of $$ \frac{8}{(t+4)^{2}} \quad(0 \leq t \leq 2) $$ inches/hour. a. Find an expression giving the total amount of rainfall after \(t\) hr. Hint: The total amount of rainfall at \(t=0\) is zero. b. How much rain had fallen after \(1 \mathrm{hr}\) ? After \(2 \mathrm{hr}\) ?