Chapter 2

In economics, total utility refers to the total satisfaction from consuming some commodity. According to the economist Samuelson: \(^{18}\) As you consume more of the same good, the total (psychological) utility increases. However, \(\ldots\) with successive new units of the good, your total utility will grow at a slower and slower rate because of a fundamental tendency for your psychological ability to appreciate more of the good to become less keen. (a) Sketch the total utility as a function of the number of units consumed. (b) In terms of derivatives, what is Samuelson saying?

Annual net sales, in billion of dollars, for the Hershey Company, the largest US producer of chocolate, is a function \(S=f(t)\) of time, \(t\), in years since 2000 . (a) Interpret the statements \(f(8)=5.1\) and \(f^{\prime}(8)=\) \(0.22\) in terms of Hershey sales. \({ }^{11}\) (b) Estimate \(f(12)\) and interpret it in terms of Hershey sales.

World meat \(^{12}\) production, \(M=f(t)\), in millions of metric tons, is a function of \(t\), years since 2000 . (a) Interpret \(f(5)=249\) and \(f^{\prime}(5)=6.5\) in terms of meat production. (b) Estimate \(f(10)\) and interpret it in terms of meat production.

(a) Let \(f(x)=\ln x .\) Use small intervals to estimate \(f^{\prime}(1), f^{\prime}(2), f^{\prime}(3), f^{\prime}(4)\), and \(f^{\prime}(5)\). (b) Use your answers to part (a) to guess a formula for the derivative of \(f(x)=\ln x\)

For some painkillers, the size of the dose, \(D\), given depends on the weight of the patient, \(W\). Thus, \(D=f(W)\), where \(D\) is in milligrams and \(W\) is in pounds. (a) Interpret the statements \(f(140)=120\) and \(f^{\prime}(140)=3\) in terms of this painkiller. (b) Use the information in the statements in part (a) to estimate \(f(145)\).

Suppose \(f(x)=\frac{1}{3} x^{3}\). Estimate \(f^{\prime}(2), f^{\prime}(3)\), and \(f^{\prime}(4)\). What do you notice? Can you guess a formula for \(f^{\prime}(x)\) ?

The quantity, \(Q \mathrm{mg}\), of nicotine in the body \(t\) minutes after a cigarette is smoked is given by \(Q=f(t)\). (a) Interpret the statements \(f(20)=0.36\) and \(f^{\prime}(20)=\) \(-0.002\) in terms of nicotine. What are the units of the numbers \(20,0.36\), and \(-0.002 ?\) (b) Use the information given in part (a) to estimate \(f(21)\) and \(f(30)\). Justify your answers.

A mutual fund is currently valued at \(\$ 80\) per share and its value per share is increasing at a rate of \(\$ 0.50\) a day. Let \(V=f(t)\) be the value of the share \(t\) days from now. (a) Express the information given about the mutual fund in term of \(f\) and \(f^{\prime}\). (b) Assuming that the rate of growth stays constant, estimate and interpret \(f(10)\).

Suppose \(C(r)\) is the total cost of paying off a car loan borrowed at an annual interest rate of \(r \%\). What are the units of \(C^{\prime}(r) ?\) What is the practical meaning of \(C^{\prime}(r) ?\) What is its sign?

A company's revenue from car sales, \(C\) (in thousands of dollars), is a function of advertising expenditure, \(a\), in thousands of dollars, so \(C=f(a)\). (a) What does the company hope is true about the sign of \(f^{\prime} ?\) (b) What does the statement \(f^{\prime}(100)=2\) mean in practical terms? How about \(f^{\prime}(100)=0.5\) ? (c) Suppose the company plans to spend about \(\$ 100,000\) on advertising. If \(f^{\prime}(100)=2\), should the company spend more or less than \(\$ 100,000\) on advertising? What if \(f^{\prime}(100)=0.5\) ?

The area of Brazil's rain forest, \(R=f(t)\), in million acres, is a function of the number of years, \(t\), since 2000 . (a) Interpret \(f(9)=740\) and \(f^{\prime}(9)=-2.7\) in terms of Brazil's rain forests. \(^{13}\) (b) Find and interpret the relative rate of change of \(f(t)\) when \(t=9\).

The number of active Facebook users hit 175 million at the end of February 2009 and 200 million \({ }^{14}\) at the end of April \(2009 .\) With \(t\) in months since the start of 2009 , let \(f(t)\) be the number of active users in millions. Estimate \(f(4)\) and \(f^{\prime}(4)\) and the relative rate of change of \(f\) at \(t=4\). Interpret your answers in terms of Facebook users.

The weight, \(w\), in kilograms, of a baby is a function \(f(t)\) of her age, \(t\), in months. (a) What does \(f(2.5)=5.67\) tell you? (b) What does \(f^{\prime}(2.5) / f(2.5)=0.13\) tell you?

Estimate the relative rate of change of \(f(t)=t^{2}\) at \(t=4\). Use \(\Delta t=0.01\).

The population, \(P\), of a city (in thousands) at time \(t\) (in years) is \(P=700 e^{0.035 t} .\) Estimate the relative rate of change of the population at \(t=3\) using (a) \(\Delta t=1\) (b) \(\Delta t=0.1\) (c) \(\Delta t=0.01\)