Chapter 5

Show that if \(f(x)\) is a positive twice-differentiable function that has a local minimum at \(x=c\), then \(g(x)=[f(x)]^{2}\) has a local minimum at \(x=c\) as well.

The growth of a particular population is described by a power law model, in which the rate of growth is given by a function: $$ r(t)=\frac{A}{(t+a)^{m}} $$ where \(A, m\), and \(a\) are all unknown constants. Given the following data for the size of the population, calculate the value for these constants that would fit the model to the data: $$ \begin{array}{ll} \hline \boldsymbol{t} & \boldsymbol{r}(\boldsymbol{t}) \\ \hline 0 & 1.89 \\ 1 & 1.31 \\ 3 & 0.988 \\ \hline \end{array} $$ Hint: Eliminate \(A\) first. It may help to then take logarithms of the equations that you derive after eliminating \(A\).

(a) Use the stability criterion to characterize the stability of the equilibria of $$x_{t+1}=\frac{10 x_{t}^{2}}{9+x_{t}^{2}}, \quad t=0,1,2, \ldots$$ (b) Use cobwebbing to decide the limit \(x_{t}\) converges to as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=3\).

Find the general antiderivative of the given function. $$ f(x)=\frac{x}{1+x} $$

Find the local maxima and minima of each of the functions. Determine whether each function has local maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. \(y=\frac{1}{3} x^{3}-x^{2}+x+1, x \in \mathbf{R}\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{x^{2}}{x^{2}+1}, x \geq 0 $$

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=3 x-5, x \in(-2,1) $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} \frac{\ln (\ln x)}{\ln x} $$

Show that if \(f(x)\) is a differentiable function with \(f(x)<0\) for all \(x \in \mathbf{R}\) and with a local maximum at \(x=c\), then \(g(x)=[f(x)]^{2}\) has a local minimum at \(x=c\).

The dynamics of a population of fish is modeled using the Beverton-Holt model: $$N_{t+1}=\frac{2 N_{t}}{1+\frac{N_{t}}{100}}$$ (a) Calculate the first ten terms of the sequence when \(N_{0}=10\). (b) Calculate the first ten terms of the sequence when \(N_{0}=150\). (c) Find all equilibria of the system, and use the stability criterion to determine which of them (if any) are stable. (d) Explain why your answers from (a) and (b) are consistent with what you have determined about the equilibria of the system.

Find the general antiderivative of the given function. $$ f(x)=5 x^{4}+\frac{5}{x^{4}} $$

Find the local maxima and minima of each of the functions. Determine whether each function has local maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. \(y=x^{2}(1-x), x \in \mathbf{R}\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\sin x, 0 \leq x \leq 2 \pi $$

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=x^{2}-2, x \in[-1,1] $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{5^{x}-1}{7^{x}-1} $$

Let $$f(x)=\frac{x}{x-1}, \quad x \neq 1 $$ (a) Show that $$ \lim _{x \rightarrow-\infty} f(x)=1 $$ and $$ \lim _{x \rightarrow+\infty} f(x)=1 $$ That is, show that \(y=1\) is a horizontal asymptote of the curve \(y=\frac{x}{x-1}\). (b) Show that $$\lim _{x \rightarrow 1^{-}} f(x)=-\infty $$ and $$\lim _{x \rightarrow 1^{+}} f(x)=+\infty$$ That is, show that \(x=1\) is a vertical asymptote of the curve \(y=\frac{x}{x-1}\) (c) Determine where \(f(x)\) is increasing and where it is decreasing. Does \(f(x)\) have local extrema? (d) Determine where \(f(x)\) is concave up and where it is concave down. Does \(f(x)\) have inflection points? (e) Sketch the graph of \(f(x)\) together with its asymptotes.

Show that if \(f(x)\) is a differentiable function for all \(x \in \mathbf{R}\) and with a local minimum at \(x=c\), then \(g(x)=\exp (-f(x))\) has a local maximum at \(x=c\).

The dynamics of a population of fish is modeled using the Beverton-Holt model: $$N_{t+1}=\frac{3 N_{t}}{1+\frac{N_{t}}{30}}$$ (a) Calculate the first ten terms of the sequence when \(N_{0}=10\). (b) Calculate the first ten terms of the sequence when \(N_{0}=120\). (c) Find all equilibria of the system, and use the stability criterion to determine which of them (if any) are stable. (d) Explain why your answers from (a) and (b) are consistent with what you have determined about the equilibria of the system.

Find the general antiderivative of the given function. $$ f(x)=x^{7}+\frac{1}{x^{7}} $$

Find the local maxima and minima of each of the functions. Determine whether each function has local maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. \(y=(x-1)^{1 / 3}, x \geq 1\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\sin \left(\pi x^{2}\right), 0 \leq x \leq 1 $$

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=(x-2)^{2}, x \in[0,3] $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{2^{x}-1}{3^{x}-1} $$

Let$$f(x)=\frac{2}{1-x^{2}}, \quad x \neq-1,1$$ (a) Show that $$\lim _{x \rightarrow+\infty} f(x)=0$$ and $$\lim _{x \rightarrow-\infty} f(x)=0$$ That is, show that \(y=0\) is a horizontal asymptote of \(f(x)\). (b) Show that $$\lim _{x \rightarrow-1^{-}} f(x)=-\infty$$ and $$\lim _{x \rightarrow-1^{+}} f(x)=+\infty$$ and that $$\lim _{x \rightarrow 1^{-}} f(x)=+\infty$$ and $$\lim _{x \rightarrow 1^{+}} f(x)=-\infty$$ That is, show that \(x=-1\) and \(x=1\) are vertical asymptotes of \(f(x)\) (c) Determine where \(f(x)\) is increasing and where it is decreasing. Does \(f(x)\) have local extrema? (d) Determine where \(f(x)\) is concave up and where it is concave down. Does \(f(x)\) have inflection points? (e) Sketch the graph of \(f(x)\) together with its asymptotes.

Optimizing Crop Yield A farmer is trying to optimize the amount of nitrogen fertilizer to add to a field. She finds that her yield per square meter increases with the amount, \(N\), of nitrogen added according to the formula: $$ Y(N)=\frac{1}{e^{-N}+1} \quad N \geq 0 $$ (a) Show that \(Y(N)\) increases monotonically with \(N\). If the cost of fertilizer is not important, this result suggests that she should add as much fertilizer as she can to the field. (b) Show either by using calculus, or by making a plot on a graphing calculator, that \(Y(N)\) is concave downward, so there are diminishing returns from using more fertilizer. (c) Suppose that the farmer includes the cost of fertilizer when determining the optimal amount to use. If the cost of one unit of fertilizer is \(C\), then her return, \(N\), becomes: $$ r(N)=Y(N)-C N=\frac{1}{e^{-N}+1}-C N \quad N \geq 0 $$ You may assume that \(C\) is a positive constant. Explain in words why, whatever the value of \(C\) is, we would expect there to be an optimal value of \(N\) that maximizes the return \(r(N)\). (d) Calculate the optimal value of \(N\) if \(C=1 / 8\) (Hint: To find when \(r^{\prime}(N)=0\), make the substitution \(u=e^{-N}\) and solve for \(u .\) You will need to use the quadratic formula.) (e) If \(C=1\) show that the optimal amount of fertilizer for the farmer to add is \(N=0\).

Consider the following discrete logistic model for the change in the size of a population over time: $$N_{t+1}=R_{0} N_{t}-\frac{1}{100} N_{t}^{2}$$ for \(t=0,1,2, \ldots\) (a) Find all equilibria when \(R_{0}=1.5\) and calculate which (if any) are stable. (b) Calculate the first ten terms of the sequence when \(N_{0}=10\) and describe what you see.

Find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+2 x} $$

Find the local maxima and minima of each of the functions. Determine whether each function has local maxima and minima and find their coordinates. For each function, find the intervals on which it is increasing and the intervals on which it is decreasing. \(y=\sqrt{1+x^{2}}, x \in \mathbf{R}\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=e^{-x}, x \in \mathbf{R} $$

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=x \ln x, x \in[1,5] $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{3^{-x}-1}{2^{x}-1} $$

Let $$f(x)=\frac{1}{x(x+1)}, \quad x \neq 0,-1$$ (a) Show that \(x=0\) and \(x=-1\) are vertical asymptotes. (b) Determine where \(f(x)\) is increasing and where it is decreasing. Does \(f(x)\) have local extrema? (c) Determine where \(f(x)\) is concave up and where it is concave down. Does \(f(x)\) have inflection points? (d) What is the behavior of the function as \(x \rightarrow \pm \infty\) ? (e) Sketch the graph of \(f(x)\) together with its asymptotes.

Ticket Price Optimization Dalmatian Airlines flies a daily flight from Los Angeles to Minneapolis. Currently they sell each ticket for \(\$ 300\), and on average 100 people take the flight, so their revenue per flight is 100 tickets \(\times \$ 300 /\) ticket \(=\$ 30,000\). They are interested in seeing whether they can increase their revenue by changing the price of a ticket. Based on market research they discover that for every \(\$ 1\) increase in ticket price, one fewer person will buy a ticket. Similarly for every \(\$ 1\) decrease in ticket price, one more person will buy a ticket. (a) What ticket price would maximize Dalmatian Airlines' revenue? (Hint: Denote the number of extra people flying on the route due to a price change by \(x\), and the cost of a ticket by \(\$ 300-x\). Then explain why the revenue to be maximized is \(R(x)=\) \((300-x)(100+x)\). You should also explain what the domain of this function is. (b) The plane can seat a maximum of 150 people. How does this information change the domain of \(R(x) ?\) What is the new optimal ticket price?

Consider the following discrete logistic model for the change in the size of a population over time: $$N_{t+1}=R_{0} N_{t}-\frac{1}{100} N_{t}^{2}$$ for \(t=0,1,2, \ldots\) (a) Find all equilibria when \(R_{0}=2.5\) and calculate which (if any) are stable. (b) Calculate the first ten terms of the sequence when \(N_{0}=10\) and describe what you see.

Find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+3 x} $$

[This problem illustrates the fact that \(f^{\prime}(c)=0\) is not a sufficient condition for the existence of a local extremum of a differentiable function.] Show that the function \(f(x)=x^{3}\) has a horizontal tangent at \(x=0 ;\) that is, show that \(f^{\prime}(0)=0\), but \(f^{\prime}(x)\) does not change sign at \(x=0\) and, hence, \(f(x)\) does not have a local extremum at \(x=0\).

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=x e^{-x}, x>0 $$

In Problems , use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=x^{2}-x, x \in[0,1] $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{2^{-x}-1}{5^{x}-1} $$

Let $$f(x)=\frac{1}{x^{2}+1}$$ (a) Show that \(y=0\) is a horizontal asymptote. (b) Does \(f(x)\) have any vertical asymptotes? (c) Follow the Steps \(1-8\) for graphing a function, to make a sketch of the graph of \(f(x)\)

Ticket Price Optimization Dalmatian Airlines also flies a daily flight from Los Angeles to Sacramento. Currently they sell each ticket for \(\$ 100\), and on average 200 people take the flight, so their revenue per flight is 200 tickets \(\times \$ 100 /\) ticket \(=\$ 20,000 .\) They are interested in seeing whether they can increase their revenue by changing the price of a ticket. Based on market research they discover that for every \(\$ 2\) increase in ticket price, one fewer person will buy a ticket. Similarly for every \(\$ 2\) decrease in ticket price, one more person will buy a ticket. (a) What ticket price would maximize Dalmatian Airlines' revenue? (Hint: Denote the number of extra people flying on the route due to a price change by \(x\), and the cost of a ticket by \(\$ 100-2 x .\) Then explain why the revenue to be maximized is \(R(x)=(100-2 x)(200+x)\). You should also explain what the domain of this function is.) (b) The plane can seat a maximum of 250 people. How does this information change the domain of \(R(x) ?\) Does this constraint affect your answer to part (a)?

Consider the following discrete logistic model for the change in the size of a population over time: $$N_{t+1}=R_{0} N_{t}-\frac{1}{100} N_{t}^{2}$$ for \(t=0,1,2, \ldots\) (a) Find all equilibria when \(R_{0}=3.5\) and calculate which (if any) are stable. (b) Calculate the first ten terms of the sequence when \(N_{0}=10\) and describe what you see.

Find the general antiderivative of the given function. $$ f(x)=e^{-3 x} $$

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=e^{-x^{2}}, x \in \mathbf{R} $$

In Problems 19-26, find c such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) $$ f(x)=x^{2} $$

Let $$f(x)=\frac{x^{2}}{1+x^{2}}, x \in \mathbf{R}$$ (a) Determine where \(f(x)\) is increasing and where it is decreasing. (b) Where is the function concave up and where is it concave down? Find all inflection points of \(f(x)\). (c) Find \(\lim _{x \rightarrow \pm \infty} f(x)\) and decide whether \(f(x)\) has a horizontal asymptote. (d) Sketch the graph of \(f(x)\) together with its asymptotes and inflection points (if they exist).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty} \frac{e^{x}-1-x}{e^{x}-x^{2}} $$

Optimal Soda Can A soda can manufacturer wants to minimize the cost of the aluminum used to make their can. The can has to hold a volume \(V\) of soda. Assuming that the thickness of the can is the same everywhere, the amount of aluminum used to make the can will be proportional to its surface area. That is, suppose the height of the can is \(h\) and the radius of the can is \(r\), as in Figure \(5.56 .\) Then the manufacturer wants to minimize: $$ S=2 \pi r h+2 \pi r^{2} $$ subject to the constraint that \(\pi r^{2} h=V .\) Here we have used the formulas for the total surface area and volume of a cylinder. (a) A real soda can has volume \(V=355 \mathrm{~cm}^{3}\) (or \(12 \mathrm{fl}\). oz.). By substituting for \(h\) in Equation \((5.17)\), write \(S\) as a function of \(r\) only. (b) Describe the behavior of \(S(r)\) as \(r \rightarrow \infty\) (c) Describe the behavior of \(S(r)\) as \(r \rightarrow 0\). (d) Based on your answers to (b) and (c), explain why you expect there to be a value of \(r\) that minimizes \(S(r) .\) Calculate this optimum radius \(r\).

A generalization of the Beverton-Holt model for population growth was created by Hassell (1975). Under Hassell's model the population \(N_{t}\) at discrete times \(t=0,1,2, \ldots\) is modeled by a recurrence equation: $$N_{t+1}=\frac{R_{0} N_{t}}{\left(1+a N_{t}\right)^{c}}$$ where \(R_{0}, a\), and \(c\) are all positive constants. (a) Explain why you would expect that \(R_{0}>1\). (b) Assume that \(c=2, R_{0}=9\), and \(a=\frac{1}{10} .\) Find all possible equilibria of the system. (c) Use the stability criterion for equilibria to determine which, if any, of the equilibria of the recursion relation are stable.

Find the general antiderivative of the given function. $$ f(x)=e^{-x / 2}-e^{-2 x} $$