Chapter 3

The cost of producing a quantity, \(q\), of a product is given by $$ C(q)=1000+30 e^{0.05 q} \text { dollars. } $$ Find the cost and the marginal cost when \(q=50\). Interpret these answers in economic terms.

The quantity, \(Q\), in tons, of material at a municipal waste site is a function of the number of years since 2000 , with $$ Q=f(t)=3 t^{2}+100 $$ Find \(f(10), f^{\prime}(10)\), and the relative rate of change \(f^{\prime} / f\) at \(t=10\). Interpret your answers in terms of waste.

The quantity, \(q\), of a certain skateboard sold depends on the selling price, \(p\), in dollars, so we write \(q=f(p)\). You are given that \(f(140)=15,000\) and \(f^{\prime}(140)=-100\). (a) What do \(f(140)=15,000\) and \(f^{\prime}(140)=-100\) tell you about the sales of skateboards? (b) The total revenue, \(R\), earned by the sale of skateboards is given by \(R=p q\). Find \(\left.\frac{d R}{d p}\right|_{p=140}\). (c) What is the sign of \(\left.\frac{d R}{d p}\right|_{p=140}\) ? If the skateboards are currently selling for \(\$ 140\), what happens to revenue if the price is increased to \(\$ 141 ?\)

With time, \(t\), in minutes, the temperature, \(H\), in degrees Celsius, of a bottle of water put in the refrigerator at \(t=0\) is given by $$ H=4+16 e^{-0.02 t} $$ How fast is the water cooling initially? After 10 minutes? Give units.

The number, \(N\), of acres of harvested land in a region is given by $$ N=f(t)=120 \sqrt{t} $$ where \(t\) is the number of years since farming began in the region. Find \(f(9), f^{\prime}(9)\), and the relative rate of change \(f^{\prime} / f\) at \(t=9\). Interpret your answers in terms of harvested land.

Show that the relative rate of change of a product \(f g\) is the sum of the relative rates of change of \(f\) and \(g\).

Carbon- 14 is a radioactive isotope used to date objects. If \(A_{0}\) represents the initial amount of carbon- 14 in the object, then the quantity remaining at time \(t\), in years, is $$ A(t)=A_{0} e^{-0.000121 t} $$ (a) A tree, originally containing 185 micrograms of carbon- 14 , is now 500 years old. At what rate is the carbon- 14 decaying? (b) In 1988 , scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained \(91 \%\) of the amount of carbon- 14 in freshly made cloth of the same material. \({ }^{6}\) According to this data, how old was the Shroud of Turin in 1988 ?

If \(f(t)=2 t^{3}-4 t^{2}+3 t-1\), find \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\).

Show that the relative rate of change of a quotient \(f / g\) is the difference between the relative rates of change of \(f\) and \(g\).

For the cost function \(C=1000+300 \ln q\) (in dollars), find the cost and the marginal cost at a production level of 500 . Interpret your answers in economic terms.

If \(f(t)=t^{4}-3 t^{2}+5 t\), find \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\).

If \(h=f^{n}\), show that $$ \frac{\left(f^{n}\right)^{\prime}}{f^{n}}=n \frac{f^{\prime}}{f} $$

In 2009 , the population, \(P\), of India was \(1.166\) billion and growing at \(1.5 \%\) annually. (a) Give a formula for \(P\) in terms of time, \(t\), measured in years since 2009 . (b) Find \(\frac{d P}{d t},\left.\frac{d P}{d t}\right|_{t=0}\), and \(\left.\frac{d P}{d t}\right|_{t=25}\). What do each of these represent in practical terms?

The time, \(\mathrm{T}\), in seconds for one complete oscillation of a pendulum is given by \(T=f(L)=1.111 \sqrt{L}\), where \(L\) is the length of the pendulum in feet. Find the following quantities, with units, and interpret in terms of the pendulum. (a) \(f(100)\) (b) \(f^{\prime}(100)\).

In 2009 , the population of Mexico was 111 million and growing \(1.13 \%\) annually, while the population of the US was 307 million and growing \(0.975 \%\) annually. \({ }^{8}\) If we measure growth rates in people/year, which population was growing faster in 2009 ?

Kleiber's Law states that the daily calorie requirement, \(C(w)\), of a mammal is proportional to the mammal's body weight \(w\) raised to the \(0.75\) power. \({ }^{1}\) If body weight is measured in pounds, the constant of proportionality is approximately 42 . (a) Give formulas for \(C(w)\) and \(C^{\prime}(w)\). (b) Find and interpret (i) \(C(10)\) and \(C^{\prime}(10)\) (ii) \(C(100)\) and \(C^{\prime}(100)\) \(\begin{array}{ll}\text { (iii) } C(1000) \text { and } C^{\prime}(1000) & 58 .\end{array}\)

(a) Find the equation of the tangent line to \(y=\ln x\) at \(x=1\) (b) Use it to calculate approximate values for \(\ln (1.1)\) and \(\ln (2)\). (c) Using a graph, explain whether the approximate values are smaller or larger than the true values. Would the same result have held if you had used the tangent line to estimate \(\ln (0.9)\) and \(\ln (0.5) ?\) Why?

Find the equation of the line tangent to the graph of \(f\) at \((1,1)\), where \(f\) is given by \(f(x)=2 x^{3}-2 x^{2}+1\).

Find the quadratic polynomial \(g(x)=a x^{2}+b x+c\) which best fits the function \(f(x)=e^{x}\) at \(x=0\), in the sense that \(g(0)=f(0)\), and \(g^{\prime}(0)=f^{\prime}(0)\), and \(g^{\prime \prime}(0)=f^{\prime \prime}(0)\) Using a computer or calculator, sketch graphs of \(f\) and \(g\) on the same axes. What do you notice?

(a) Find the equation of the tangent line to \(f(x)=x^{3}\) at the point where \(x=2\). (b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of the function, will the estimates be overestimates or underestimates?

Find the equation of the line tangent to the graph of \(f(t)=6 t-t^{2}\) at \(t=4\). Sketch the graph of \(f(t)\) and the tangent line on the same axes.

If you are outdoors, the wind may make it feel a lot colder than the thermometer reads. You feel the windchill temperature, which, if the air temperature is \(20^{\circ} \mathrm{F}\), is given in \({ }^{\circ} \mathrm{F}\) by \(W(v)=48.17-27.2 v^{0.16}\), where \(v\) is the wind velocity in mph for \(5 \leq v \leq 60^{2}\) (a) If the air temperature is \(20^{\circ} \mathrm{F}\), and the wind is blowing at \(40 \mathrm{mph}\), what is the windchill temperature, to the nearest degree? (b) Find \(W^{\prime}(40)\), and explain what this means in terms of windchill.

(a) Use the formula for the area of a circle of radius \(r\), \(A=\pi r^{2}\), to find \(d A / d r\) (b) The result from part (a) should look familiar. What does \(d A / d r\) represent geometrically? (c) Use the difference quotient to explain the observation you made in part (b).

Suppose \(W\) is proportional to \(r^{3}\). The derivative \(d W / d r\) is proportional to what power of \(r\) ?

The cost to produce \(q\) items is \(C(q)=1000+2 q^{2}\) dollars. Find the marginal cost of producing the \(25^{\text {th }}\) item. Interpret your answer in terms of costs.

The demand curve for a product is given by \(q=300-3 p\), where \(p\) is the price of the product and \(q\) is the quantity that consumers buy at this price. (a) Write the revenue as a function, \(R(p)\), of price. (b) Find \(R^{\prime}(10)\) and interpret your answer in terms of revenue. (c) For what prices is \(R^{\prime}(p)\) positive? For what prices is it negative?

The yield, \(Y\), of an apple orchard (measured in bushels of apples per acre) is a function of the amount \(x\) of fertilizer in pounds used per acre. Suppose $$ Y=f(x)=320+140 x-10 x^{2} $$ (a) What is the yield if 5 pounds of fertilizer is used per acre? (b) Find \(f^{\prime}(5)\). Give units with your answer and interpret it in terms of apples and fertilizer. (c) Given your answer to part (b), should more or less fertilizer be used? Explain.

The demand for a product is given, for \(p, q \geq 0\), by $$ p=f(q)=50-0.03 q^{2} $$ (a) Find the \(p\) - and \(q\) -intercepts for this function and interpret them in terms of demand for this product. (b) Find \(f(20)\) and give units with your answer. Explain what it tells you in terms of demand. (c) Find \(f^{\prime}(20)\) and give units with your answer. \(\mathrm{Ex}\) plain what it tells you in terms of demand.

The cost (in dollars) of producing \(q\) items is given by \(C(q)=0.08 q^{3}+75 q+1000\) (a) Find the marginal cost function. (b) Find \(C(50)\) and \(C^{\prime}(50)\). Give units with your answers and explain what each is telling you about costs of production.

A ball is dropped from the top of the Empire State Building. The height, \(y\), of the ball above the ground (in feet) is given as a function of time, \(t\), (in seconds) by $$ y=1250-16 t^{2} $$ (a) Find the velocity of the ball at time \(t\). What is the sign of the velocity? Why is this to be expected? (b) When does the ball hit the ground, and how fast is it going at that time? Give your answer in feet per second and in miles per hour ( \(1 \mathrm{ft} / \mathrm{sec}=15 / 22 \mathrm{mph}\) ).

Let \(f(x)=x^{3}-6 x^{2}-15 x+20\). Find \(f^{\prime}(x)\) and all values of \(x\) for which \(f^{\prime}(x)=0 .\) Explain the relationship between these values of \(x\) and the graph of \(f(x)\).

Show that for any power function \(f(x)=x^{n}\), we have \(f^{\prime}(1)=n\)

If the demand curve is a line, we can write \(p=b+m q\), where \(p\) is the price of the product, \(q\) is the quantity sold at that price, and \(b\) and \(m\) are constants. (a) Write the revenue as a function of quantity sold. (b) Find the marginal revenue function.