Chapter 3

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

The depth of the water, \(y\), in meters, in the Bay of Fundy, Canada, is given as a function of time, \(t\), in hours after midnight, by the function $$ y=10+7.5 \cos (0.507 t) $$ How quickly is the tide rising or falling (in meters/hour) at each of the following times? (a) \(6: 00 \mathrm{am}\) (b) \(9: 00 \mathrm{am}\) (c) Noon (d) \(6: 00 \mathrm{pm}\)

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ w=\frac{3 y+y^{2}}{5+y} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(y=x^{2}+4 x-3 \ln x\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=\sqrt{x}(x+1)$$

The average adult takes about 12 breaths per minute. As a patient inhales, the volume of air in the lung increases. As the patient exhales, the volume of air in the lung decreases. For \(t\) in seconds since start of the breathing cycle, the volume of air inhaled or exhaled since \(t=0\) is given \(^{9}\), in hundreds of cubic centimeters, by $$ A(t)=-2 \cos \left(\frac{2 \pi}{5} t\right)+2 $$ (a) How long is one breathing cycle? (b) Find \(A^{\prime}(1)\) and explain what it means.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=\frac{1+z}{\ln z} $$

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(f(t)=A e^{t}+B \ln t\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(\theta)=\theta\left(\theta^{-1 / 2}-\theta^{-2}\right)$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=\frac{a x+b}{c x+k} $$

For \(f(t)=4-2 e^{t}\), find \(f^{\prime}(-1), f^{\prime}(0)\), and \(f^{\prime}(1)\). Graph \(f(t)\), and draw tangent lines at \(t=-1, t=0\), and \(t=1 .\) Do the slopes of the lines match the derivatives you found?

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=k x^{2}$$

Paris, France, has a latitude of approximately \(49^{\circ} \mathrm{N}\). If \(t\) is the number of days since the start of 2009 , the number of hours of daylight in Paris can be approximated by $$ D(t)=4 \cos \left(\frac{2 \pi}{365}(t-172)\right)+12 $$ (a) Find \(D(40)\) and \(D^{\prime}(40) .\) Explain what this tells about daylight in Paris. (b) Find \(D(172)\) and \(D^{\prime}(172)\). Explain what this tells about daylight in Paris.

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=\left(a x^{2}+b\right)^{3} $$

Find the equation of the tangent line to the graph of \(y=3^{x}\) at \(x=1\). Check your work by sketching a graph of the function and the tangent line on the same axes.

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=a x^{2}+b x+c$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(x)=a x e^{-b x} $$

Find the equation of the tangent line to \(y=e^{-2 t}\) at \(t=0 .\) Check by sketching the graphs of \(y=e^{-2 t}\) and the tangent line on the same axes.

Find the derivative. Assume \(a, b, c, k\) are constants. $$Q=a P^{2}+b P^{3}$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ f(t)=a e^{b t} $$

Find the equation of the tangent line to \(f(x)=10 e^{-0.2 x}\) at \(x=4\).

Find the derivative. Assume \(a, b, c, k\) are constants. $$v=a t^{2}+\frac{b}{t^{2}}$$

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ g(\alpha)=e^{\alpha e^{-2 \alpha}} $$

A fish population is approximated by \(P(t)=10 e^{0.6 t}\), where \(t\) is in months. Calculate and use units to explain what each of the following tells us about the population: (a) \(P(12)\) (b) \(P^{\prime}(12)\)

Find the derivative. Assume \(a, b, c, k\) are constants. $$P=a+b \sqrt{t}$$

If \(f(x)=(3 x+8)(2 x-5)\), find \(f^{\prime}(x)\) and \(f^{\prime \prime}(x)\).

The world's population is about \(f(t)=6.8 e^{0.012 t}\) billion, \(^{3}\) where \(t\) is time in years since \(2009 .\) Find \(f(0)\), \(f^{\prime}(0), f(10)\), and \(f^{\prime}(10)\). Using units, interpret your answers in terms of population.

Find the derivative. Assume \(a, b, c, k\) are constants. $$V=\frac{4}{3} \pi r^{2} b$$

Find the equation of the tangent line to the graph of \(f(x)=x^{2} e^{-x}\) at \(x=0\). Check by graphing this function and the tangent line on the same axes.

The demand curve for a product is given by $$ q=f(p)=10,000 e^{-0.25 p} $$ where \(q\) is the quantity sold and \(p\) is the price of the product, in dollars. Find \(f(2)\) and \(f^{\prime}(2)\). Explain in economic terms what information each of these answers gives you.

Find the derivative. Assume \(a, b, c, k\) are constants. $$w=3 a b^{2} q$$

Find the equation of the tangent line to the graph of \(f(x)=\frac{2 x-5}{1}\) at the point at which \(x=0\).

Worldwide production of solar power, in megawatts, can be modeled by \(f(t)=1040(1.3)^{t}\), where \(t\) is years \(^{4}\) since \(2000 .\) Find \(f(0), f^{\prime}(0), f(15)\), and \(f^{\prime}(15) .\) Give units and interpret your answers in terms of solar power.

Find the derivative. Assume \(a, b, c, k\) are constants. $$h(x)=\frac{a x+b}{c}$$

The quantity of a drug, \(Q \mathrm{mg}\), present in the body \(t\) hours after an injection of the drug is given is $$ Q=f(t)=100 t e^{-0.5 t} $$ Find \(f(1), f^{\prime}(1), f(5)\), and \(f^{\prime}(5)\). Give units and interpret the answers.

A new DVD is available for sale in a store one week after its release. The cumulative revenue, \(\$ R\), from sales of the DVD in this store in week \(t\) after its release is $$ R=f(t)=350 \ln t \quad \text { with } t>1 $$ Find \(f(5), f^{\prime}(5)\), and the relative rate of change \(f^{\prime} / f\) at \(t=5 .\) Interpret your answers in terms of revenue.

A drug concentration curve is given by \(C=f(t)=\) \(20 t e^{-0.04 t}\), with \(C\) in \(\mathrm{mg} / \mathrm{ml}\) and \(t\) in minutes. (a) Graph \(C\) against \(t\). Is \(f^{\prime}(15)\) positive or negative? Is \(f^{\prime}(45)\) positive or negative? Explain. (b) Find \(f(30)\) and \(f^{\prime}(30)\) analytically. Interpret them in terms of the concentration of the drug in the body.

In 2009 , the population of Hungary \(^{5}\) was approximated by $$ P=9.906(0.997)^{t} $$ where \(P\) is in millions and \(t\) is in years since 2009 . Assume the trend continues. (a) What does this model predict for the population of Hungary in the year \(2020 ?\) (b) How fast (in people/year) does this model predict Hungary's population will be decreasing in \(2020 ?\)

Let \(f(x)=x^{3}-4 x^{2}+7 x-11 .\) Find \(f^{\prime}(0), f^{\prime}(2)\), \(f^{\prime}(-1) .\)

For positive constants \(c\) and \(k\), the Monod growth curve describes the growth of a population, \(P\), as a function of the available quantity of a resource, \(r\) : $$ P=\frac{c r}{k+r} . $$ Find \(d P / d r\) and interpret it in terms of the growth of the population.

With \(t\) in years since January 1,2010 , the population \(P\) of Slim Chance is predicted by $$ P=35,000(0.98)^{t} $$ At what rate will the population be changing on January \(1,2023 ?\)

Let \(f(t)=t^{2}-4 t+5\) (a) Find \(f^{\prime}(t)\). (b) Find \(f^{\prime}(1)\) and \(f^{\prime}(2)\). (c) Use a graph of \(f(t)\) to check that your answers to part (b) are reasonable. Explain.

If \(p\) is price in dollars and \(q\) is quantity, demand for a product is given by $$ q=5000 e^{-0.08 p} $$ (a) What quantity is sold at a price of \(\$ 10\) ? (b) Find the derivative of demand with respect to price when the price is \(\$ 10\) and interpret your answer in terms of demand.

Some antique furniture increased very rapidly in price over the past decade. For example, the price of a particular rocking chair is well approximated by $$ V=75(1.35)^{t} $$ where \(V\) is in dollars and \(t\) is in years since 2000 . Find the rate, in dollars per year, at which the price is increasing at time \(t\)

Find the rate of change of a population of size \(P(t)=\) \(t^{3}+4 t+1\) at time \(t=2\)

The height of a sand dune (in centimeters) is represented by \(f(t)=700-3 t^{2}\), where \(t\) is measured in years since 2005\. Find \(f(5)\) and \(f^{\prime}(5)\). Using units, explain what each means in terms of the sand dune.

The quantity demanded of a certain product, \(q\), is given in terms of \(p\), the price, by $$ q=1000 e^{-0.02 p} $$ (a) Write revenue, \(R\), as a function of price. (b) Find the rate of change of revenue with respect to price. (c) Find the revenue and rate of change of revenue with respect to price when the price is \(\$ 10\). Interpret your answers in economic terms.

At a time \(t\) hours after it was administered, the concentration of a drug in the body is \(f(t)=27 e^{-0.14 t} \mathrm{ng} / \mathrm{ml}\). What is the concentration 4 hours after it was administered? At what rate is the concentration changing at that time?

Zebra mussels are freshwater shellfish that first appeared in the St. Lawrence River in the early \(1980 \mathrm{~s}\) and have spread throughout the Great Lakes. Suppose that \(t\) months after they appeared in a small bay, the number of zebra mussels is given by \(Z(t)=300 t^{2} .\) How many zebra mussels are in the bay after four months? At what rate is the population growing at that time? Give units.

If \(\frac{d}{d t}(t f(t))=1+f(t)\), what is \(f^{\prime}(t) ?\)