Chapter 14

Find \(y^{\prime}\). $$ y=2 \ln 5 x $$

If \(f(x)=\frac{3 \ln \sqrt{x}}{x}\), what is \(f^{\prime}(e)\) ?

For Problems 1 through 3, find \(\frac{d y}{d x}\). \(y=x^{2} \cdot 2^{x}\)

Find \(y^{\prime}\). $$ y=\pi \ln \sqrt{x} $$

For Problems 2 through 5, compute \(y^{\prime} .\) $$ y=\ln \left(3 x^{2}\right)\left(\text { Hint }: \text { write } \ln \left(3 x^{2}\right) \text { as } \ln 3+2 \ln x .\right) $$

Find \(\frac{d y}{d x}\). \(y=\frac{5 \cdot 2^{x}}{3}\)

Find \(y^{\prime}\). $$ y=\frac{\ln 3 x}{5} $$

Find \(\frac{d y}{d x}\). \(y=\frac{x^{5} 5^{x}}{5}\)

Find \(y^{\prime}\). $$ y=x \ln x $$

Compute \(y^{\prime} .\) $$ y=\frac{\ln x}{x} $$

Find \(y^{\prime}\). $$ y=\frac{\ln \sqrt{2 x}}{x} $$

Compute \(y^{\prime} .\) $$ y=x \ln \left(\frac{1}{x}\right) $$

Find \(f^{\prime}(x)\) if (a) \(f(x)=x^{2}+e^{x}+x^{e}+e^{2}\). (b) \(f(x)=\pi e^{x}-\frac{6 e^{x}}{\sqrt{29}}\). (c) \(f(x)=3 e^{x+3}\). (Hint: Break this up into the product of \(e^{x}\) and a constant.)

Find \(y^{\prime}\). $$ y=3 \log x $$

For Problems 6 through 13, differentiate the given function. $$ f(x)=x \ln \left(\frac{1}{x}\right) $$

Using the definition of derivative, show that the derivative of \(a^{x}\) is \(a^{x}\) times the slope of the tangent to \(a^{x}\) at \(x=0\). (We've done this, but refresh your memory.)

Find \(y^{\prime}\). $$ y=\frac{\log _{2} x}{3} $$

Differentiate the given function. $$ f(x)=\left(\frac{3 \ln \left(3^{6} x^{7}\right)}{\pi}\right)+\frac{3 \ln \left(3^{6}\right)}{\pi} $$

Show that \(f(x)=\frac{\ln \sqrt{3 x}}{2}+3\) is invertible. Find \(f^{-1}(x)\).

Differentiate the given function. $$ f(x)=e^{5 x} \ln \left(\frac{\pi}{\sqrt{x}}\right) $$

Graph \(f(x)=e^{x}-x\). Only use a calculator to check your work after working on your own. (a) Find \(f^{\prime}(x)\). Draw a number line and indicate where \(f^{\prime}\) is positive, zero, and negative. (b) Label the \(x\) - and \(y\) -coordinates of any local extrema (local maxima or minima). (c) Using your picture, determine how many solutions there are to the following equations. i. \(f(x)=5\) ii. \(f(x)=0.5\) Notice that these equations are "intractable" - try to solve \(e^{x}-x=5\) algebraically to see what this means. If we want to estimate the solutions, we can do so using a graphing calculator. At this point, we should know how many solutions to expect.

Show that \(g(x)=\pi \log _{2}(\pi x)-\pi^{2}\) is invertible. Find \(g^{-1}(x)\).

Differentiate the given function. $$ f(x)=3^{x}(\log x) $$

Find the quantity indicated. (a) \(y=\ln 5 x+\ln x^{5}+\ln 5^{x}\) i. Find \(y^{\prime}\). ii. Find the slope of the graph of the function at \(x=1\). (b) \(f(x)=\log _{10} x\) i. Find \(f^{\prime}(x)\). ii. Find \(f^{\prime}(100)\). (c) \(P(x)=7^{x}\) i. Find \(P^{\prime}(x)\). ii. Find the instantaneous rate of change of \(P\) with respect to \(x\) when \(x=0\). (d) \(y=e^{3 x}\) i. Find \(y^{\prime}\). ii. Find the slope of the graph of the function at \(x=0\). (e) \(f(x)=14 e^{x / 2}\) i. Find \(f^{\prime}\). ii. Find \(f^{\prime}(\ln 9)\) and simplify your answer.

Find and classify the critical points of \(f(x)=x \ln x\).

Differentiate the given function. $$ f(x)=\frac{\ln \left(2 x^{3}\right)}{3 e^{x}} $$

Differentiate \(y=e^{3 x} \ln \left(\frac{1}{\sqrt{5 x}}\right)\)

What is the lowest value taken on by the function \(g(x)=x^{2} \ln x ?\) Is there a highest value? Explain.

Differentiate the given function. $$ f(x)=\frac{x+\ln \left(\frac{1}{x}\right)}{x^{2}}(\text { Conserve your energy. }) $$

Use a tangent line approximation of \(\ln x\) at \(x=1\) to approximate: (a) \(\ln (0.9)\). (b) \(\ln (1.1)\)

Differentiate the given function. $$ f(x)=x^{\pi}+\pi^{x}+\ln \left(\frac{\pi}{x}\right) $$

Graph \(f(x)=\sqrt{x}-\ln x\), indicating all local maxima, minima, and points of inflection. Do this without your graphing calculator. (You can use your calculator to check your answer.) To aid in doing the graphing, do the following. (a) On a number line, indicate the sign of \(f^{\prime}\). Above this number line draw arrows indicating whether \(f\) is increasing or decreasing. (b) On a number line indicate the sign of \(f^{\prime \prime}\). Above this number line indicate the concavity of \(f\). (c) Find \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow \infty} f(x)\) using all tools available to you. You should be able to give a strong argument supporting your answer to the former. The latter requires a bit more ingenuity, but you can do it.

Differentiate the given function. $$ f(x)=x^{2} \ln \left(x \sqrt{\frac{61}{2 x}}\right) $$

Let \(f(x)=\ln x-x\). (a) What is the domain of this function? (b) Find all the critical points of \(f\). (The critical points must be in the domain of \(f .\) ) (c) By looking at the sign of \(f^{\prime}\), find all local maxima and minima. Give both the \(x\) and \(y\) -coordinates of the extrema. (d) Find \(f^{\prime \prime}\). Where is \(f\) concave up and where is \(f\) concave down? (e) Sketch the graph of \(\ln x-x\) without using a calculator (except possibly to check your work).

Using what you know about the graph of \(\ln x\), sketch the graphs of the following. (a) \(y=|\ln x|\) (b) \(y=\ln (|x|)\) (c) \(y=|\ln (|x|)|\) (d) For parts (a), (b), and (c), locate all critical points and identify all local maxima and minima.

Graph \(f(x)=\ln x+1 / x\) for \(x>0\), indicating all local maxima and minima and any points of inflection. While answering this problem, do the following. (a) On a number line indicate the sign of \(f^{\prime}\). Above this number line draw arrows indicating whether \(f\) is increasing or \(f\) is decreasing. (b) On a number line indicate the sign of \(f^{\prime \prime}\). Above this number line write " \(f\) is concave up" and " \(f\) is concave down" as appropriate. (c) Using all tools available, guess \(\lim _{x \rightarrow 0^{+}}\left(\ln x+\frac{1}{x}\right)\). Then make a convincing argument supporting your answer. (Note: Usually when graphing we would first look at \(f\) itself and determine where that is positive and negative if that information is easy to obtain. In this problem we didn't ask you to do that only because the information is not particularly easy to obtain. Now that we're done, we do in fact have that sign information.)

Differentiate. Spend time writing each of these in a form that makes the differentiation easy. (a) \(y=3 \ln 5 x+6 \ln \left(\frac{3}{x}\right)\) (b) \(y=20 \log \left(\frac{x}{100}\right)\) (c) \(y=\frac{k e^{2 k x}}{\sqrt{k+1}}\) (d) \(y=\left(\frac{(\ln 2) e^{5 x}}{\ln 4}\right)+\frac{(\ln 2) e^{\ln 2}}{\ln 3}\)

A report from the United Nations Food and Agriculture Organization ( Boston Globe, July 22,1996 ) made several projections about the growth of the world's population and need for food over the 55 -year time period from 1995 to 2050 . Among them were the following. I. The world's population is predicted to grow from \(5.7\) billion to \(9.8\) billion. II. North America will need to increase food production by a total of \(30 \%\) to satisfy the demands of its population. III. Africa will need to increase food production by a total of \(300 \%\) to satisfy the demands of its population. IV. The report also notes that sharing the world's food resources more fairly would "probably eliminate most cases of undernourishment." Given these projections, answer the following questions: (a) By what total percentage is the world's population predicted to increase over these 55 years? (b) Assuming that population growth is exponential, find an equation for the world's population as a function of time, with \(t=0\) in 1995 . What is the annual percentage growth rate? (c) According to your equation, what is the instantaneous rate of growth at \(t=0 ?\) At \(t=55 ?\) (d) What is the average rate of growth between \(t=0\) and \(t=55\) ? (e) In what year is the instantaneous rate of growth equal to the average rate of growth? Use a sketch to illustrate whether this point should occur before or after \(t=27.5\) years. (f) Assuming the food production grows linearly, write an equation for North America's annual food production as a function of time, where \(N_{0}\) is the amount produced in \(1995(t=0)\). Do the same for Africa, where \(A_{0}\) is the 1995 amount.

In a 1960 s program to bring exotic animal species to the United States, 60 oryx (a 700 -pound antelope with sharp 3-foot-long horns) were brought to the deserts of New Mexico. Thirty years later, the oryx population in New Mexico had grown to 2000 and was "destroying natural habitat." (Boston Globe, July 31, 1996.) (a) Assuming that the growth of the population was exponential, write an equation for \(P(t)\), the number of ory \(x\) as a function of time, letting \(t=0\) when they were first imported. (b) What was their annual growth rate? (c) What was the doubling time for the population? (d) According to your equation, what was the instantaneous rate of growth at \(t=0 ?\) At \(t=30 ?\) (e) What was the average rate of change over the 30 -year period? (f) Interpret each of the following in words. i. \(P(10)\) ii. \(P^{-1}(200)\) (g) Estimate \(P^{-1}(200)\).

A radioactive substance decays exponentially. Suppose its half-life is 5000 years and the initial amount of radioactive substance is denoted by \(R_{0}\). (a) Write an equation of the form \(R(t)=R_{0} e^{k t}\) for \(R(t)\), the amount of radioactive material left after \(t\) years. (b) If \(R_{0}=3000 \mathrm{mg}\), at what rate is the radioactive substance decaying at time \(t=0\) ?

Suppose we know a population grows exponentially; \(P(2)=1000\) and \(P(4)=1300\). Find the growth equation. (Hint: Write \(P=P_{0} e^{k t}\), or some other form of exponential growth. Put in the given information. Since you don't know \(P_{0}\), divide one equation by the other so that the \(P_{0}\) 's cancel.)

Suppose you put \(\$ 500\) in a bank account and your balance grows exponentially according to the equation $$ M=500 e^{0.08 t}, $$ where \(M=M(t)=\) the amount of money in the account at time \(t\). (a) Write the growth equation for the amount of money in the account in the form \(M=500 A^{t}\) (b) What is the annual growth rate of the money in the account? (Banks refer to this as the effective annual yield.) Please give your answer to the nearest tenth of a percent. (c) What is the instantaneous rate of change of money with respect to time? (d) When will you have enough money to buy a round-the-world plane ticket costing \(\$ 1599 ?\)

Suppose you put \(M_{0}\) dollars in a bank account with \(6 \%\) interest compounded annually. (a) Write an equation for \(M(t)\), the amount of money in the account at time \(t, t\) measured in years. Construct a continuous model. (b) Find \(M^{\prime}(t)\), the instantaneous rate of change of money with respect to time.

In early summer the fly population in Maine grows exponentially. The population at any time \(t\) can be given by \(P(t)=P_{0} e^{k t}\) for some constant \(k\), where \(t\) is measured in days. Suppose that at some date, which we will designate as \(t=0\), there are 200 flies. Thirty days later there are 900 flies. (a) Find the constant \(k\). (b) The mosquito population is also growing exponentially. At time \(t=0\) there are 100 mosquitoes, and the mosquito population doubles every 10 days. Write a function \(M(t)\) that gives the number of mosquitoes at time \(t\). (c) When will the number of flies and the number of mosquitoes be equal? (d) Find \(P^{\prime}(t)\). (e) Find \(M^{\prime}(t)\). (f) Find the rate at which each of the populations is growing when the populations are the same size. Which is growing more rapidly?

Suppose a population grows exponentially according to the equation $$ P=P_{0} e^{0.4 t}. $$ (a) Write the growth equation for the population in the form \(P=P_{0} A^{t}\). (b) What is the annual growth rate of the population? (c) What is the instantaneous rate of change of population with respect to time? (d) How long does it take the population to double?

Let \(f(x)=\frac{2 \ln x}{x}\). Does \(f\) have global extrema? Find the absolute maximum and minimum values taken on by \(f\) if these values exist. (In order to complete your argument, you will need to compute a limit. Make this argument explicit.)

Let \(g(x)=x^{2} \cdot 2^{x}\). Find all local extrema. Does \(g(x)\) have a global maximum? A global minimum? If so, where? Explain your reasoning carefully.

Analyze the critical points of \(h(x)=x^{3} \cdot 3^{x}\). What is the absolute minimum value of \(h\) ?