Chapter 2

The function \(C(q)\) gives the cost in dollars to produce \(q\) barrels of olive oil. (a) What are the units of marginal cost? (b) What is the practical meaning of the statement \(M C=3\) for \(q=100 ?\)

It costs \(\$ 4800\) to produce 1295 items and it costs \(\$ 4830\) to produce 1305 items. What is the approximate marginal cost at a production level of 1300 items?

Write the Leibniz notation for the derivative of the given function and include units. The distance to the ground, \(D\), in feet, of a skydiver is a function of the time \(t\) in minutes since the skydiver jumped out of the airplane.

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=3 t^{2}\). (a) Find the average velocity between \(t=1\) and \(t=\) \(1+h\) if: (i) \(h=0.1\), (ii) \(h=0.01\), (iii) \(h=\underline{0.001 \text { . }}\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=1\).

The cost, \(C=f(w)\), in dollars of buying a chemical is a function of the weight bought, \(w\), in pounds. (a) In the statement \(f(12)=5\), what are the units of the 12 ? What are the units of the 5 ? Explain what this is saying about the cost of buying the chemical. (b) Do you expect the derivative \(f^{t}\) to be positive or negative? Why? (c) In the statement \(f^{\prime}(12)=0.4\), what are the units of the 12 ? What are the units of the \(0.4\) ? Explain what this is saying about the cost of buying the chemical.

The distance (in feet) of an object from a point is given by \(s(t)=t^{2}\), where time \(t\) is in seconds. (a) What is the average velocity of the object between \(t=3\) and \(t=5 ?\) (b) By using smaller and smaller intervals around 3 , estimate the instantaneous velocity at time \(t=3\).

For \(q\) units of a product, a manufacturer's cost is \(C(q)\) dollars and revenue is \(R(q)\) dollars, with \(C(500)=\) \(7200, R(500)=9400, M C(500)=15\), and \(M R(500)=20\) (a) What is the profit or loss at \(q=500 ?\) (b) If production is increased from 500 to 501 units, by approximately how much does profit change?

The time for a chemical reaction, \(T\) (in minutes), is a function of the amount of catalyst present, \(a\) (in milliliters), so \(T=f(a)\). (a) If \(f(5)=18\), what are the units of 5 ? What are the units of 18 ? What does this statement tell us about the reaction? (b) If \(f^{\prime}(5)=-3\), what are the units of 5 ? What are the units of \(-3\) ? What does this statement tell us?

An economist is interested in how the price of a certain item affects its sales. At a price of \(\$ p\), a quantity, \(q\), of the item is sold. If \(q=f(p)\), explain the meaning of each of the following statements: (a) \(f(150)=2000\) (b) \(f^{\prime}(150)=-25\)

The size, \(S\), of a tumor (in cubic millimeters) is given by \(S=2^{t}\), where \(t\) is the number of months since the tumor was discovered. Give units with your answers. (a) What is the total change in the size of the tumor during the first six months? (b) What is the average rate of change in the size of the tumor during the first six months? (c) Estimate the rate at which the tumor is growing at \(t=6\). (Use smaller and smaller intervals.)

Let \(C(q)\) represent the total cost of producing \(q\) items. Suppose \(C(15)=2300\) and \(C^{\prime}(15)=108\). Estimate the total cost of producing: (a) 16 items (b) 14 items.

Graph the functions described in parts (a)-(d). (a) First and second derivatives everywhere positive. (b) Second derivative everywhere negative; first derivative everywhere positive. (c) Second derivative everywhere positive; first derivative everywhere negative. (d) First and second derivatives everywhere negative.

The temperature, \(T\), in degrees Fahrenheit, of a cold yam placed in a hot oven is given by \(T=f(t)\), where \(t\) is the time in minutes since the yam was put in the oven. (a) What is the sign of \(f^{\prime}(t) ?\) Why? (b) What are the units of \(f^{\prime}(20)\) ? What is the practical meaning of the statement \(f^{\prime}(20)=2 ?\)

Let \(f(x)=5^{x}\). Use a small interval to estimate \(f^{\prime}(2)\). Now improve your accuracy by estimating \(f^{\prime}(2)\) again, using an even smaller interval.

To produce 1000 items, the total cost is \(\$ 5000\) and the marginal cost is \(\$ 25\) per item. Estimate the costs of producing 1001 items, 999 items, and 1100 items.

In Problems 10-11, use the values given for each function. (a) Does the derivative of the function appear to be positive or negative over the given interval? Explain. (b) Does the second derivative of the function appear to be positive or negative over the given interval? Explain. $$ \begin{array}{c|c|c|c|c|c} \hline t & 100 & 110 & 120 & 130 & 140 \\ \hline w(t) & 10.7 & 6.3 & 4.2 & 3.5 & 3.3 \\ \hline \end{array} $$

On May 9,2007, CBS Evening News had a \(4.3\) point rating. (Ratings measure the number of viewers.) News executives estimated that a \(0.1\) drop in the ratings for the CBS Evening News corresponds to a \(\$ 5.5\) million drop in revenue. \({ }^{9}\) Express this information as a derivative. Specify the function, the variables, the units, and the point at which the derivative is evaluated.

(a) Let \(g(t)=(0.8)^{t}\). Use a graph to determine whether \(g^{\prime}(2)\) is positive, negative, or zero. (b) Use a small interval to estimate \(g^{\prime}(2)\).

Let \(C(q)\) represent the cost and \(R(q)\) represent the revenue, in dollars, of producing \(q\) items. (a) If \(C(50)=4300\) and \(C^{\prime}(50)=24\), estimate \(C(52)\). (b) If \(C^{\prime}(50)=24\) and \(R^{\prime}(50)=35\), approximately how much profit is earned by the \(51^{\text {st }}\) item? (c) If \(C^{\prime}(100)=38\) and \(R^{\prime}(100)=35\), should the company produce the \(101^{\text {st }}\) item? Why or why not?

Use the values given for each function. (a) Does the derivative of the function appear to be positive or negative over the given interval? Explain. (b) Does the second derivative of the function appear to be positive or negative over the given interval? Explain. $$ \begin{array}{c|c|c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline s(t) & 12 & 14 & 17 & 20 & 31 & 55 \\ \hline \end{array} $$

Sketch the graph of a function whose first derivative is everywhere negative and whose second derivative is positive for some \(x\) -values and negative for other \(x\) -values.

Meteorologists define the temperature lapse rate to be \(-d T / d z\) where \(T\) is the air temperature in Celsius at altitude \(z\) kilometers above the ground. (a) What are the units of the lapse rate? (b) What is the practical meaning of a lapse rate of \(6.5 ?\)

Estimate \(f^{\prime}(2)\) for \(f(x)=3^{x}\). Explain your reasoning.

A company's cost of producing \(q\) liters of a chemical is \(C(q)\) dollars; this quantity can be sold for \(R(q)\) dollars. Suppose \(C(2000)=5930\) and \(R(2000)=7780\). (a) What is the profit at a production level of 2000 ? (b) If \(M C(2000)=2.1\) and \(M R(2000)=2.5\), what is the approximate change in profit if \(q\) is increased from 2000 to \(2001 ?\) Should the company increase or decrease production from \(q=2000 ?\) (c) If \(M C(2000)=4.77\) and \(M R(2000)=4.32\), should the company increase or decrease production from \(q=2000 ?\)

IBM-Peru uses second derivatives to assess the relative success of various advertising campaigns. They assume that all campaigns produce some increase in sales. If a graph of sales against time shows a positive second derivative during a new advertising campaign, what does this suggest to IBM management? Why? What does a negative second derivative suggest?

Investing \(\$ 1000\) at an annual interest rate of \(r \%\), compounded continuously, for 10 years gives you a balance of \(\$ B\), where \(B=g(r)\). Give a financial interpretation of the statements: (a) \(g(5) \approx 1649 .\) (b) \(g^{\prime}(5) \approx 165\). What are the units of \(g^{\prime}(5)\) ?

An industrial production process costs \(C(q)\) million dollars to produce \(q\) million units; these units then sell for \(R(q)\) million dollars. If \(C(2.1)=5.1, R(2.1)=6.9\) \(M C(2.1)=0.6\), and \(M R(2.1)=0.7\), calculate (a) The profit earned by producing \(2.1\) million units (b) The approximate change in revenue if production increases from \(2.1\) to \(2.14\) million units. (c) The approximate change in revenue if production decreases from \(2.1\) to \(2.05\) million units. (d) The approximate change in profit in parts (b) and (c).

Let \(f(x)\) be the elevation in feet of the Mississippi River \(x\) miles from its source. What are the units of \(f^{\prime}(x)\) ? What can you say about the sign of \(f^{\prime}(x) ?\)

The average weight, \(W\), in pounds, of an adult is a function, \(W=f(c)\), of the average number of Calories per day, \(c\), consumed. (a) Interpret the statements \(f(1800)=155\) and \(f^{\prime}(2000)=0\) in terms of diet and weight. (b) What are the units of \(f^{\prime}(c)=d W / d c ?\)

(a) Graph \(f(x)=x^{2}\) and \(g(x)=x^{2}+3\) on the same axes. What can you say about the slopes of the tangent lines to the two graphs at the point \(x=0 ?\) \(x=1 ? x=2 ? x=a\), where \(a\) is any value? (b) Explain why adding a constant to any function will not change the value of the derivative at any point.

The cost, \(C\) (in dollars), to produce \(g\) gallons of a chemical can be expressed as \(C=f(g)\). Using units, explain the meaning of the following statements in terms of the chemical: (a) \(f(200)=1300\) (b) \(f^{\prime}(200)=6\)

The weight, \(W\), in lbs, of a child is a function of its age, \(a\), in years, so \(W=f(a)\). (a) Do you expect \(f^{\prime}(a)\) to be positive or negative? Why? (b) What does \(f(8)=45\) tell you? Give units for the numbers 8 and 45 . (c) What are the units of \(f^{\prime}(a)\) ? Explain what \(f^{\prime}(a)\) tells you in terms of age and weight. (d) What does \(f^{\prime}(8)=4\) tell you about age and weight? (e) As \(a\) increases, do you expect \(f^{\prime}(a)\) to increase or decrease? Explain.

Estimate \(P^{\prime}(0)\) if \(P(t)=200(1.05)^{t} .\) Explain how you obtained your answer.

Sketch a graph of a continuous function \(f\) with the following properties: \- \(f^{\prime}(x)>0\) for all \(x\) \- \(f^{\prime \prime}(x)<0\) for \(x<2\) and \(f^{\prime \prime}(x)>0\) for \(x>2\).

Let \(G\) be annual US government purchases, \(T\) be annual US tax revenues, and \(Y\) be annual US output of all goods and services. All three quantities are given in dollars. Interpret the statements about the two derivatives, called fiscal policy multipliers. (a) \(d Y / d G=0.60\) (b) \(d Y / d T=-0.26\)

A recent study reports that men who retired late developed Alzheimer's at a later stage than those who stopped work earlier. Each additional year of employment was associated with about a six-week later age of onset. Express these results as a statement about the derivative of a function. State clearly what function you use, including the units of the dependent and independent variables.

A city grew in population throughout the 1980 s. The population was at its largest in 1990 , and then shrank throughout the 1990 s. Let \(P=f(t)\) represent the population of the city \(t\) years since \(1980 .\) Sketch graphs of \(f(t)\) and \(f^{\prime}(t)\), labeling the units on the axes.

For three minutes the temperature of a feverish person has had positive first derivative and negative second derivative. Which of the following is correct? (a) The temperature rose in the last minute more than it rose in the minute before. (b) The temperature rose in the last minute, but less than it rose in the minute before. (c) The temperature fell in the last minute but less than it fell in the minute before. (d) The temperature rose two minutes ago but fell in the last minute.

The thickness, \(P\), in \(\mathrm{mm}\), of pelican eggshells depends on the concentration, \(c\), of \(\mathrm{PCBs}\) in the eggshell, measured in ppm (parts per million); that is, \(P=f(c)\). (a) The derivative \(f^{\prime}(c)\) is negative. What does this tell you? (b) Give units and interpret \(f(200)=0.28\) and \(f^{\prime}(200)=-0.0005\) in terms of \(\mathrm{PCBs}\) and eggs.

Yesterday's temperature at \(t\) hours past midnight was \(f(t){ }^{\circ} \mathrm{C}\). At noon the temperature was \(20^{\circ} \mathrm{C}\). The first derivative, \(f^{\prime}(t)\), decreased all morning, reaching a low of \(2^{\circ} \mathrm{C} /\) hour at noon, then increased for the rest of the day. Which one of the following must be correct? (a) The temperature fell in the morning and rose in the afternoon. (b) At 1 pm the temperature was \(18^{\circ} \mathrm{C}\). (c) At I pm the temperature was \(22^{\circ} \mathrm{C}\). (d) The temperature was lower at noon than at any other time. (e) The temperature rose all day.

Sketch the graph of a function \(f\) such that \(f(2)=5\), \(f^{\prime}(2)=1 / 2\), and \(f^{\prime \prime}(2)>0 .\)

Estimate the instantaneous rate of change of the function \(f(x)=x \ln x\) at \(x=1\) and at \(x=2 .\) What do these values suggest about the concavity of the graph between \(\mathrm{J}\) and \(2 ?\)

A function \(f\) has \(f(5)=20, f^{\prime}(5)=2\), and \(f^{\prime \prime}(x)<0\), for \(x \geq 5 .\) Which of the following are possible values for \(f(7)\) and which are impossible? (a) 26 (b) 24 (c) 22

On October 17,2006, in an article called "US Population Reaches 300 Million," the \(\mathrm{BBC}\) reported that the US gains 1 person every 11 seconds. If \(f(t)\) is the US population in millions \(t\) years after October 17,2006, find \(f(0)\) and \(f^{\prime}(0)\).

"Winning the war on poverty" has been described cynically as slowing the rate at which people are slipping below the poverty line. Assuming that this is happening: (a) Graph the total number of people in poverty against time. (b) If \(N\) is the number of people below the poverty line at time \(t\), what are the signs of \(d N / d t\) and \(d^{2} N / d t^{2} ?\) Explain.

Suppose that \(f(t)\) is a function with \(f(25)=3.6\) and \(f^{\prime}(25)=-0.2\). Estimate \(f(26)\) and \(f(30)\).

Draw a possible graph of \(y=f(x)\) given the following information about its derivative. \- \(f^{\prime}(x)>0\) for \(x<-1\) \- \(f^{\prime}(x)<0\) for \(x>-1\) \- \(f^{\prime}(x)=0\) at \(x=-1\)

Let \(P(t)\) represent the price of a share of stock of a corporation at time \(t\). What does each of the following statements tell us about the signs of the first and second derivatives of \(P(t) ?\) (a) "The price of the stock is rising faster and faster." (b) "The price of the stock is close to bottoming out."

Suppose that \(f(x)\) is a function with \(f(20)=345\) and \(f^{\prime}(20)=6\). Estimate \(f(22)\).

Draw the graph of a continuous function \(y=f(x)\) that satisfies the following three conditions: \- \(f^{\prime}(x)>0\) for \(13\) \- \(f^{\prime}(x)=0\) at \(x=1\) and \(x=3\)