Chapter 10

The concentration (in milligrams/cubic centimeter) 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 horizontal asymptote of \(C(t)\). b. Interpret your result.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=-x^{2}+2 x+4 $$

Find the relative maxima and relative minima, if any, of each function. $$ F(x)=\frac{1}{3} x^{3}-x^{2}-3 x+4 $$

The number of major crimes committed in the city of 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 \(t(t=0\) corresponds to 2000 ). Enraged by the dramatic increase in the crime rate, the citizens of Bronxville, with the help of the local police, organized "Neighborhood Crime Watch" groups in early 2004 to combat this menace. Show that the growth in the crime rate was maximal in 2005 , giving credence to the claim that the Neighborhood Crime Watch program was working.

Certain proteins, known as enzymes, serve as catalysts for chemical reactions in living things. In 1913 Leonor Michaelis and L. M. Menten discovered the following formula giving the initial speed \(V\) (in moles/liter/second) at which the reaction begins in terms of the amount of substrate \(x\) (the substance that is being acted upon, measured in moles/liter): $$ V=\frac{a x}{x+b} $$ where \(a\) and \(b\) are positive constants. a. Find the horizontal asymptote of \(V\). b. What does the result of part (a) tell you about the initial speed at which the reaction begins, if the amount of substrate is very large?

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(x)=2 x^{2}+3 x+7 $$

Find the relative maxima and relative minima, if any, of each function. $$ F(t)=3 t^{5}-20 t^{3}+20 $$

The percentage of foreign-born residents in the United States from 1910 through 2000 is approximated by the function \(P(t)=0.04363 t^{3}-0.267 t^{2}-1.59 t+14.7 \quad(0 \leq t \leq 9)\) where \(t\) is measured in decades, with \(t=0\) corresponding to \(1910 .\) Show that the percentage of foreign-born residents was lowest in early 1970 . Hint: Use the quadratic formula.

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 2000 . Sketch the graph of the function \(G\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=2 x^{3}+1 $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=x^{4}-4 x^{3}+8 $$

In a study conducted at the National Institute of Mental Health, researchers followed the development of the cortex, the thinking part of the brain, in 307 children. Using repeated magnetic resonance imaging scans from childhood to the latter teens, they measured the thickness (in millimeters) of the cortex of children of age \(t\) yr with the highest IQs-121 to 149 . These data lead to the model $$ \begin{array}{r} S(t)=0.000989 t^{3}-0.0486 t^{2}+0.7116 t+1.46 \\ (5 \leq t \leq 19) \end{array} $$ Show that the cortex of children with superior intelligence reaches maximum thickness around age 11 . Hint: Use the quadratic formula.

The number of major crimes per 100,000 committed in a city between 2000 and 2007 is approximated by the function $$ N(t)=-0.1 t^{3}+1.5 t^{2}+80 \quad(0 \leq t \leq 7) $$ where \(N(t)\) denotes the number of crimes per 100,000 committed in year \(t\), with \(t=0\) corresponding to 2000 . Enraged by the dramatic increase in the crime rate, the citizens, with the help of the local police, organized Neighborhood Crime Watch groups in early 2004 to combat this menace. Sketch the graph of the function \(N^{\prime}\) and interpret your results. Is the Neighborhood Crime Watch program working?

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(x)=x^{3}-6 x $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=3 x^{4}-2 x^{3}+4 $$

Refer to Exercise 66. The researchers at the Institute also measured the thickness (also in millimeters) of the cortex of children of age \(t\) yr who were of average intelligence. These data lead to the model $$ \begin{array}{r} A(t)=-0.00005 t^{3}-0.000826 t^{2}+0.0153 t+4.55 \\ (5 \leq t \leq 19) \end{array} $$ Show that the cortex of children with average intelligence reaches maximum thickness at age \(6 \mathrm{yr}\).

An efficiency study showed that the total number of cordless telephones assembled by an average worker at Delphi Electronics \(t\) hr after starting work at 8 a.m. is given by $$ N(t)=-\frac{1}{2} t^{3}+3 t^{2}+10 t \quad(0 \leq t \leq 4) $$ Sketch the graph of the function \(N\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=\frac{1}{3} x^{3}-2 x^{2}-5 x-10 $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=\frac{x+1}{x} $$

The average annual price of single-family homes in Massachusetts between 1990 and 2002 is approximated by the function \(P(t)=-0.183 t^{3}+4.65 t^{2}-17.3 t+200 \quad(0 \leq t \leq 12)\) where \(P(t)\) is measured in thousands of dollars and \(t\) is measured in years, with \(t=0\) corresponding to 1990 . In what year was the average annual price of single-family homes in Massachusetts lowest? What was the approximate lowest average annual price? Hint: Use the quadratic formula.

The concentration (in millimeters/cubic centimeter) of a certain drug in a patient's bloodstream \(t\) hr after injection is given by $$ C(t)=\frac{0.2 t}{t^{2}+1} $$ Sketch the graph of the function \(C\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=2 x^{3}+3 x^{2}-12 x-4 $$

Find the relative maxima and relative minima, if any, of each function. $$ h(x)=\frac{x}{x+1} $$

After the economy softened, the sky-high office space rents of the late 1990 s started to come down to earth. The function \(R\) gives the approximate price per square foot in dollars, \(R(t)\), of prime space in Boston's Back Bay and Financial District from \(1997(t=0)\) through 2002, where \(R(t)=-0.711 t^{3}+3.76 t^{2}+0.2 t+36.5 \quad(0 \leq t \leq 5)\) Show that the office space rents peaked at about the middle of 2000 . What was the highest office space rent during the period in question? Hint: Use the quadratic formula.

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. Sketch the graph of the function \(T\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(t)=t+\frac{9}{t} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x+\frac{9}{x}+2 $$

The total world population is forecast to be \(P(t)=0.00074 t^{3}-0.0704 t^{2}+0.89 t+6.04 \quad(0 \leq t \leq 10)\) in year \(t\), where \(t\) is measured in decades with \(t=0\) corresponding to 2000 and \(P(t)\) is measured in billions. a. Show that the world population is forecast to peak around 2071 . Hint: Use the quadratic formula. b. What will the population peak at?

When organic waste is dumped into a pond, the oxidation process that takes place reduces the pond's oxygen content. However, given time, nature will restore the oxygen content to its natural level. Suppose the oxygen content \(t\) days after organic waste has been dumped into the pond is given by $$ f(t)=100\left(\frac{t^{2}-4 t+4}{t^{2}+4}\right) \quad(0 \leq t<\infty) $$ percent of its normal level. Sketch the graph of the function \(f\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(t)=2 t+\frac{3}{t} $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=2 x^{2}+\frac{4000}{x}+10 $$

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=\frac{x}{1-x} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=\frac{x}{1+x^{2}} $$

It has been conjectured that 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

The speed of traffic flow in miles per hour on a stretch of Route 123 between 6 a.m. and 10 a.m. on a typical workday is approximated by the function $$ f(t)=20 t-40 \sqrt{t}+52 \quad(0 \leq t \leq 4) $$ where \(t\) is measured in hours, with \(t=0\) corresponding to 6 a.m. Sketch the graph of \(f\) and interpret your results.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(x)=\frac{2 x}{x^{2}+1} $$

Find the relative maxima and relative minima, if any, of each function. $$ g(x)=\frac{x}{x^{2}-1} $$

During a flu epidemic, the total number of students on a state university campus who had contracted influenza by the \(x\) th day was given by $$ N(x)=\frac{3000}{1+99 e^{-x}} \quad(x \geq 0) $$ a. How many students had influenza initially? b. Derive an expression for the rate at which the disease was being spread and prove that the function \(N\) is increasing on the interval \((0, \infty)\). c. Sketch the graph of \(N\). What was the total number of students who contracted influenza during that particular epidemic?

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ f(t)=t^{2}-\frac{16}{t} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x e^{-x} $$

A liquid carries a drug into an organ of volume \(V \mathrm{~cm}^{3}\) at the rate of \(a \mathrm{~cm}^{3} / \mathrm{sec}\) and leaves at the same rate. The concentration of the drug in the entering liquid is \(c \mathrm{~g} / \mathrm{cm}^{3}\). Letting \(x(t)\) denote the concentration of the drug in the organ at any time \(t\), we have \(x(t)=c\left(1-e^{-a t / V}\right)\). a. Show that \(x\) is an increasing function on \((0, \infty)\). b. Sketch the graph of \(x\).

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(x)=x^{2}+\frac{2}{x} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{2} e^{-x} $$

Suppose the source of current in an electric circuit is a battery. Then the power output \(P\) (in watts) obtained if the circuit has a resistance of \(R\) ohms is given by $$ P=\frac{E^{2} R}{(R+r)^{2}} $$ where \(E\) is the electromotive force in volts and \(r\) is the internal resistance of the battery in ohms. If \(E\) and \(r\) are constant, find the value of \(R\) that will result in the greatest power output. What is the maximum power output?

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(s)=\frac{s}{1+s^{2}} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x-\ln x $$

In deep water, a wave of length \(L\) travels with a velocity $$ v=k \sqrt{\frac{L}{C}+\frac{C}{L}} $$ where \(k\) and \(C\) are positive constants. Find the length of the wave that has a minimum velocity.

Find the relative extrema, if any, of each function. Use the second derivative test, if applicable. $$ g(x)=\frac{1}{1+x^{2}} $$

Find the relative maxima and relative minima, if any, of each function. $$ f(x)=x^{2} \ln x $$

In an autocatalytic chemical reaction, the product formed acts as a catalyst for the reaction. If \(Q\) is the amount of the original substrate present initially and \(x\) is the amount of catalyst formed, then the rate of change of the chemical reaction with respect to the amount of catalyst present in the reaction is $$ R(x)=k x(Q-x) \quad(0 \leq x \leq Q) $$ where \(k\) is a constant. Show that the rate of the chemical reaction is greatest at the point when exactly half of the original substrate has been transformed.