Chapter 2

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.

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 cost, \(C,\) of a steak, in dollars, is a function of the weight, \(W,\) of the steak, in pounds.

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 \(q=100 ?\)

Write the Leibniz notation for the derivative of the given function and include units. The number, \(N,\) of gallons of gas left in a gas tank is a function of the distance, \(D,\) in miles, the car has been driven.

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 position \(s\) of a car at time \(t\) is given in the following table. $$\begin{array}{c|c|c|c|c|c|c}\hline t(\mathrm{sec}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\\\\hline s(\mathrm{ft}) & 0 & 0.5 & 1.8 & 3.8 & 6.5 & 9.6 \\\\\hline\end{array}$$ (a) Find the average velocity over the interval \(0 \leq t \leq\) 0.2. (b) Find the average velocity over the interval \(0.2 \leq t \leq\) 0.4. (c) Use the previous answers to estimate the instantaneous velocity of the car at \(t=0.2\).

Write the Leibniz notation for the derivative of the given function and include units. An employee's pay, \(P\), in dollars, for a week is a function of the number of hours worked, \(H\)

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=4 t^{2}+3\). (a) Find the average velocity between \(t=1\) and \(t=\) \(1+h\) if: (i) \(\quad h=0.1\) (ii) \(\quad h=0.01,\) (iii) \(\quad h=0.001\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=1\).

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) \(\quad f(150)=2000\) (b) \(\quad f^{\prime}(150)=-25\)

For \(q\) units of a product, a manufacturer's cost is \(C(q)\) dollars and revenue is \(R(q)\) dollars, with \(C(500)=\) \(7200, \quad R(500)=9400, \quad M C(500) \quad=\quad 15, \quad\) 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 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 \(t=2\) and \(t=5 ?\) (b) By using smaller and smaller intervals around \(2,\) estimate the instantaneous velocity at time \(t=2\)

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 \(f^{\prime}\) 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 cost of recycling \(q\) tons of paper is given in the fol. lowing table. Estimate the marginal cost at \(q=2000\) Give units and interpret your answer in terms of cost. At approximately what production level does marginal cost appear smallest?$$\begin{array}{c|c|c|c|c|c|c} \hline q \text { (tons) } & 1000 & 1500 & 2000 & 2500 & 3000 & 3500 \\ \hline C(q) \text { (dollars) } & 2500 & 3200 & 3640 & 3825 & 3900 & 4400 \\ \hline \end{array}$$

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.)

When you breathe, a muscle (called the diaphragm) reduces the pressure around your lungs and they expand to fill with air. The table shows the volume of a lung as a function of the reduction in pressure from the diaphragm. Pulmonologists (lung doctors) define the compliance of the lung as the derivative of this function. 10 (a) What are the units of compliance? (b) Estimate the maximum compliance of the lung. (c) Explain why the compliance gets small when the lung is nearly full (around 1 liter). $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Pressure reduction } \\ \text { (cm of water) } \end{array} & \begin{array}{c} \text { Volume } \\ \text { (iters) } \end{array} \\ \hline 0 & 0.20 \\ \hline 5 & 0.29 \\ \hline 10 & 0.49 \\ \hline 15 & 0.70 \\ \hline 20 & 0.86 \\ \hline 25 & 0.95 \\ \hline 30 & 1.00 \\ \hline \end{array}$$

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.

Let \(g(x)=4^{x} .\) Use small intervals to estimate \(g^{\prime}(1)\).

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.

A yam has just been taken out of the oven and is cooling off before being eaten. The temperature, \(T,\) of the yam (measured in degrees Fahrenheit) is a function of how long it has been out of the oven, \(t\) (measured in minutes). Thus, we have \(T=f(t)\) (a) Is \(f^{\prime}(t)\) positive or negative? Why? (b) What are the units for \(f^{\prime}(t) ?\)

Let \(C(q)\) represent the cost and \(R(q)\) represent the revenue, in dollars, of producing \(q\) items. 5 (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?

Using slopes to left and right of \(0,\) estimate \(R^{\prime}(0)\) if \(R(x)=100(1.1)^{x}\)

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 \(f^{\prime}(x) ?\)

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. $$\begin{array}{c|c|c|c|c|c}\hline t & 100 & 110 & 120 & 130 & 140 \\\\\hline w(t) & 10.7 & 6.3 & 4.2 & 3.5 & 3.3 \\\\\hline\end{array}$$

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 \(6.5 ?\)

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).

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. $$\begin{array}{c|c|c|c|c|c|c}\hline t & 0 & 1 & 2 & 3 & 4 & 5 \\\\\hline s(t) & 12 & 14 & 17 & 20 & 31 & 55 \\\\\hline\end{array}$$

For the function \(f(x)=3^{x},\) estimate \(f^{\prime}(1) .\) From the graph of \(f(x),\) would you expect your estimate to be greater than or less than the true value of \(f^{\prime}(1) ?\)

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 \(g^{\prime}(5) ?\)

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 \(q=2000 ?\)

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.

Table 2.4 gives \(P=f(t),\) the number of households, in millions, in the US with cable television \(t\) years since \(1998^{4}\) (a) Does \(f^{\prime}(4)\) appear to be positive or negative? What does this tell you about the percent of households with cable television? (b) Estimate \(f^{\prime}(2) .\) Estimate \(f^{\prime}(10) .\) Explain what each is telling you, in terms of cable television. $$\begin{array}{c|c|c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\\hline P & 64.65 & 66.25 & 66.732 & 65.727 & 65.141 & 64.874 & 60.958 \\\\\hline\end{array}$$

Table 2.13 shows the cost. \(C(q),\) and revenue, \(R(q),\) in terms of quantity \(q .\) Estimate the marginal cost. \(C^{\prime}(q)\) and marginal revenue, \(R^{\prime}(q),\) for \(q\) between 0 and 7. $$\begin{array}{r|r|r|r|r|r|r|r|r}\hline q & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\hline C(q) & 9 & 10 & 12 & 15 & 19 & 24 & 30 & 37 \\\\\hline R(q) & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 \\\\\hline \end{array}$$

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?

The following table gives the percent of the US population living in urban areas as a function of year. $$\begin{array}{c|c|c|c|c|c}\hline \text { Year } & 1800 & 1830 & 1860 & 1890 & 1920 \\\\\hline \text { Percent } & 6.0 & 9.0 & 19.8 & 35.1 & 51.2 \\\\\hline \text { Year } & 1950 & 1980 & 1990 & 2000 & 2005 \\\\\hline \text { Percent } & 64.0 & 73.7 & 75.2 & 79.0 & 79.0 \\\\\hline\end{array}$$ (a) Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990 (b) Estimate the rate at which this percent is increasing at the year 1990 (c) Estimate the rate of change of this function for the year 1830 and explain what it is telling you.

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 \(f^{\prime}(c)=d W / d c ?\)

Table 2.14 shows the cost, \(C(q),\) and revenue, \(R(q),\) in terms of quantity \(q .\) Estimate the marginal cost. \(M C(q)\) and marginal revenue, \(M R(q),\) for \(q\) between 0 and 6, $$\begin{array}{r|r|r|r|r|r|r}\hline q & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline C(q) & 20 & 60 & 120 & 200 & 300 & 420 \\\\\hline R(q) & 100 & 220 & 330 & 410 & 450 & 480 \\\\\hline\end{array}$$

Values of \(f(t)\) are given in the following table. (a) Does this function appear to have a positive or negative first derivative? Second derivative? Explain. (b) Estimate \(f^{\prime}(2)\) and \(f^{\prime}(8)\) $$\begin{array}{c|c|c|c|c|c|c}\hline t & 0 & 2 & 4 & 6 & 8 & 10 \\\\\hline f(t) & 150 & 145 & 137 & 122 & 98 & 56 \\\\\hline\end{array}$$

(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) \(\quad f(200)=1300\) (b) \(\quad f^{\prime}(200)=6\)

The table gives the number of passenger cars, \(C=f(t)\) in millions, \(^{24}\) in the US in the year \(t\) (a) Do \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\) appear to be positive or negative during the period \(1975-1990 ?\) (b) Do \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\) appear to be positive or negative during the period \(1990-2000 ?\) (c) Estimate \(f^{\prime}(2005) .\) Using units, interpret your answer in terms of passenger cars. $$\begin{array}{c|c|c|c|c|c|c|c}\hline t & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\ \hline C & 106.7 & 121.6 & 127.9 & 133.7 & 128.4 & 133.6 & 136.6 \\\\\hline\end{array}$$

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) \(\quad d Y / d G=0.60\) (b) \(\quad d Y / d T=-0.26\)

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

The weight, \(W,\) in 1 bs, 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.

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.

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 eggshells.

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 for two minutes but fell in the last minute.

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 1 and \(2 ?\)

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