Chapter 5

Determine whether each function has absolute 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. $$ y=\sqrt{1+x^{2}}, x \in \mathbf{R} $$

Suppose that the size of a fish population at generation \(t\) is given by $$ N_{t+1}=1.5 N_{t} e^{-0.001 N_{t}} $$ for \(t=0,1,2, \ldots\) (a) Assume that \(N_{0}=100\). Find the size of the fish population at generation \(t\) for \(t=1,2, \ldots, 20\) (b) Assume that \(N_{0}=800\). Find the size of the fish population at generation \(t\) for \(t=1,2, \ldots, 20\) (c) Determine all fixed points. On the basis of your computations in (a) and (b), make a guess as to what will happen to the population in the long run, starting from (i) \(N_{0}=100\) and (ii) \(N_{0}=800\). (d) Use the cobwebbing method to illustrate your answer in (a). (e) Explain why the dynamical system converges to the nontrivial fixed point.

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\cos \left[\pi\left(x^{2}-1\right)\right], 2 \leq x \leq 3 $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x^{7}+\frac{1}{x^{7}} $$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=(x-2)^{2}, x \in[0,3]\)

. 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 \(x=c\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{3^{-x}-1}{2^{x}-1} $$

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 \(x=0\).

Suppose that the size of a fish population at generation \(t\) is given by $$ N_{t+1}=10 N_{t} e^{-0.01 N_{t}} $$ for \(t=0,1,2, \ldots\) (a) Assume that \(N_{0}=100\). Find the size of the fish population at generation \(t\) for \(t=1,2, \ldots, 20\). (b) Show that if \(N_{0}=100 \ln 10\), then \(N_{t}=100 \ln 10\) for \(t=\) \(1,2,3, \ldots ;\) that is, show that \(N^{*}=100 \ln 10\) is a nontrivial fixed point, or equilibrium. How would you find \(N^{*}\) ? Are there any other equilibria? (c) On the basis of your computations in (a), make a prediction about the long-term behavior of the fish population when \(N_{0}=\) \(100 .\) How does your answer compare with that in (b)? (d) Use the cobwebbing method to illustrate your answer in (c). In Problems \(18-20\), consider the following discrete-time dynamical system, which is called the discrete logistic model and which models the size of a population over time: $$ N_{t+1}=N_{t}\left[1+R\left(1-\frac{N_{t}}{100}\right)\right] $$ for \(t=0,1,2, \ldots\)

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=e^{x}, x \in \mathbf{R} $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+2 x} $$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=-x^{2}+1, x \in[-2,1]\)

Find the dimensions of a right circular cylindrical can (with bottom and top closed) that has a volume of 1 liter and that minimizes the amount of material used. (Note: One liter corresponds to \(1000 \mathrm{~cm}^{3}\).)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{2^{-x}-1}{5^{x}-1} $$

Suppose that \(f(x)\) is twice differentiable on \(\mathbf{R}\), with \(f(x)>0\) for \(x \in \mathbf{R}\). Show that if \(f(x)\) has a local maximum at \(x=c\), then \(g(x)=\ln f(x)\) also has a local maximum at \(x=c\).

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\ln x, x>0 $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{1}{1+3 x} $$

Use a graphing calculator to determine all local and global extrema of the functions on their respective domains. \(f(x)=x^{2}-x, x \in[0,1]\)

Find the dimensions of a right circular cylinder that is open on the top, is closed on the bottom, holds 1 liter, and uses the least amount of material.

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1-x}{x^{2}} $$

Determine all inflection points. $$ f(x)=x^{3}-2, x \in \mathbf{R} $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=e^{-x^{2} / 2}, x \in \mathbf{R} $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=e^{-3 x} $$

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

A circular sector with radius \(r\) and angle \(\theta\) has area \(A .\) Find \(r\) and \(\theta\) so that the perimeter is smallest when (a) \(A=2\) and (b) \(A=10 .\) (Note: \(A=\frac{1}{2} r^{2} \theta\), and the length of the arc \(s=r \theta\), when \(\theta\) is measured in radians; see Figure 5.59.)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1-x-\frac{x^{2}}{2}}{x^{3}} $$

Determine all inflection points. $$ f(x)=(x-3)^{5}, 0 \in \mathbf{R} $$

Determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. $$ y=\frac{1}{1+e^{-x}}, x \in \mathbf{R} $$

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=e^{x / 2}+e^{-x / 2} $$

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=(x-4)^{2}\)

A circular sector with radius \(r\) and angle \(\theta\) has area \(A .\) Find \(r\) and \(\theta\) so that the perimeter is smallest for a given area \(A .(\) Note: \(A=\frac{1}{2} r^{2} \theta\), and the length of the arc \(s=r \theta\), when \(\theta\) is measured in radians; see Figure \(5.59 .\) )

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{(\ln x)^{2}}{x^{2}} $$

Determine all inflection points. $$ f(x)=e^{-x^{2}}, x \geq 0 $$

Investigate the canonical discrete-time logistic growth model $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ Show that for \(r>1\), there are two fixed points. For which values of \(r\) is the nonzero fixed point locally stable?

Sketch the graph of (a) a function that is increasing at an accelerating rate; and (b) a function that is increasing at a decelerating rate. (c) Assume that your functions in (a) and (b) are twice differentiable. Explain in each case how you could check the respective properties by using the first and the second derivatives. Which of the functions is concave up, and which is concave down?

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2 e^{2 x} $$

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=-x^{2}\)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} \frac{x^{7}}{e^{x}} $$

Determine all inflection points. $$ f(x)=x e^{-x}, x \geq 0 $$

Investigate the canonical discrete-time logistic growth model $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ Use a calculator or a spreadsheet to simulate the canonical discrete-time logistic growth model with \(x_{0}=0.1\) for \(t=0,1,2, \ldots, 100\), and describe the behavior when (a) \(r=3.20\) (b) \(r=3.52\) (c) \(r=3.80\) (d) \(r=3.83\) (e) \(r=3.828\)

Show that if \(f(x)\) is the linear function \(y=m x+b\), then increases in \(f(x)\) are proportional to increases in \(x .\) That is, if we increase \(x\) by \(\Delta x\), then \(f(x)\) increases by the same amount \(\Delta y\), regardless of the value of \(x .\) Compute \(\Delta y\) as a function of \(\Delta x\).

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=-3 e^{-4 x} $$

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=-(x+3)^{2}\)

Find two positive numbers \(a\) and \(b\) such that \(a+b=20\) and \(a b\) is a maximum.

Use l'Hospital's rule to find the limits. $$ \lim _{\rightarrow(\pi / 2)-} \frac{\tan x}{\sec ^{2} x} $$

Determine all inflection points. $$ f(x)=\tan x,-\frac{\pi}{2}

Consider density-dependent population growth models of the form $$ N_{t+1}=R\left(N_{t}\right) N_{t} $$ The function \(R(N)\) describes the per capita growth. Various forms have been considered. For each function \(R(N)\), find all nontrivial fixed points \(N^{*}\) (i.e, \(N^{*}>0\) ) and determine the stability as a function of the parameter values. We assume that the function parameters are \(r>0, K>0\), and \(\gamma>1 .\) for \(t=0,1,2, \ldots\) \(R(N)=r N^{1-\gamma}\)

We frequently must solve equations of the form \(f(x)=\) When \(f\) is a continuous function on \([a, b]\) and \(f(a)\) and \(f(b\) have opposite signs, the intermediate-value theorem guarantee that there exists at least one solution of the equation \(f(x)=0\) \([a, b]\) (a) Explain in words why there exists exactly one solution \((a, b)\) if, in addition, \(f\) is differentiable in \((a, b)\) and \(f^{\prime}(x)\) is eithe strictly positive or strictly negative throughout \((a, b)\). (b) Use the result in (a) to show that $$ x^{3}-4 x+1=0 $$ has exactly one solution in \([-1,1]\).

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\frac{1}{e^{2 x}} $$

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=x^{3}\)