Chapter 5

. Find two numbers \(a\) and \(b\) such that \(a-b=4\) and \(a b\) is a minimum.

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

Determine all inflection points. $$ f(x)=\ln x+\frac{1}{x}, x>0 $$

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 \(t=0,1,2, \ldots\) \(R(N)=\frac{r}{1+N / K}\)

Suppose that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Show that if \(f^{\prime}(x)<0\) for all \(x \in(a, b)\), then \(f\) is decreasing on \([a, b] .\)

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

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=x^{5}\)

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

This problem illustrates the fact that \(f^{\prime \prime}(c)=0\) is not a sufficient condition for an inflection point of a twice-differentiable function.] Show that the function \(f(x)=x^{4}\) has \(f^{\prime \prime}(0)=0\) but that \(f^{\prime \prime}(x)\) does not change sign at \(x=0\) and, hence, \(f(x)\) does not have an inflection point at \(x=0\).

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 \(t=0,1,2, \ldots\) \(R(N)=e^{r(1-N / K)}\)

Suppose that \(f\) is twice differentiable on an open interval \(I\). Show that if \(f^{\prime \prime}(x)<0\), then \(f\) is concave down.

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

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=(x+1)^{3}\)

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

Logistic Equation Suppose that the size of a population at time \(t\) is denoted by \(N(t)\) and satisfies $$ N(t)=\frac{100}{1+3 e^{-2 t}} $$ for \(t \geq 0\) (a) Show that \(N(0)=25\). (b) Show that \(N(t)\) is strictly increasing. (c) Show that $$ \lim _{t \rightarrow \infty} N(t)=100 $$ (d) Show that \(N(t)\) has an inflection point when \(N(t)=50-\) that is, when the size of the population is at half its limiting value. (e) Use your results in (a)-(d) to sketch the graph of \(N(t)\).

Suppose the size of a population at time \(t\) is \(N(t)\), and the growth rate of the population is given by the logistic growth function $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. (a) Graph the growth rate \(\frac{d N}{d t}\) as a function of \(N\) for \(r=3\) and \(K=10\) (b) The function \(f(N)=r N(1-N / K), N \geq 0\), is differentiable for \(N>0\). Compute \(f^{\prime}(N)\), and determine where the function \(f(N)\) is increasing and where it is decreasing.

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

Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=-(x-3)^{5}\)

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} x^{5} e^{-x} $$

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=\frac{2}{3} x^{3}-2 x^{2}-6 x+2 \text { for }-2 \leq x \leq 5 $$

Suppose that the size of a population at time \(t\) is \(N(t)\) and the growth rate of the population is given by the logistic growth function $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. The per capita growth rate is defined by $$ g(N)=\frac{1}{N} \frac{d N}{d t}=r\left(1-\frac{N}{K}\right) $$ (a) Graph \(g(N)\) as a function of \(N\) for \(N \geq 0\) when \(r=3\) and \(K=10\) (b) The function \(g(N)=r(1-N / K), N \geq 0\), is differentiable for \(N>0\). Compute \(g^{\prime}(N)\), and determine where the function \(g(N)\) is increasing and where it is decreasing.

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

Show that \(f(x)=|x|\) has a local minimum at \(x=0\) but \(f(x)\) is not differentiable at \(x=0\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} x^{n} e^{-x}, n \in \mathbf{N} $$

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=x^{4}-2 x^{2}, x \in \mathbf{R} $$

The growth rate of a plant depends on the amount of resources available. A simple and frequently used model for resource-dependent growth is the Monod model, according to which the growth rate is equal to $$ f(R)=\frac{a R}{k+R}, R \geq 0 $$ where \(R\) denotes the resource level and \(a\) and \(k\) are positive constants. When is the growth rate increasing? When is it decreasing?

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\cos \left(\frac{x}{5}\right)-\sin \left(\frac{x}{5}\right) $$

Show that \(f(x)=|x-1|\) has a local minimum at \(x=1\) but \(f(x)\) is not differentiable at \(x=1\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}} \sqrt{x} \ln x $$

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=\left|x^{2}-9\right|,-4 \leq x \leq 5 $$

Suppose that the growth rate of a population is given by $$ f(N)=N\left(1-\left(\frac{N}{K}\right)^{\theta}\right) $$ where \(N\) is the size of the population, \(K\) is a positive constant denoting the carrying capacity, and \(\theta\) is a parameter greater than 1\. Find \(f^{\prime}(N)\), and determine where the growth rate is increasing and where it is decreasing.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=2 \sin \left(\frac{\pi}{2} x\right)-3 \cos \left(\frac{\pi}{2} x\right) $$

Show that \(f(x)=\left|x^{2}-1\right|\) has local minima at \(x=1\) and \(x=-1\) but \(f(x)\) is not differentiable at \(x=1\) or \(x=-1\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}} x^{2} \ln x $$

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=\sqrt{|x|}, x \in \mathbf{R} $$

Spruce budworms are a major pest that defoliates balsam fir. They are preyed upon by birds. A model for the per capita predation rate is given by $$ f(N)=\frac{a N}{k^{2}+N^{2}} $$ where \(N\) denotes the density of spruce budworms and \(a\) and \(k\) are positive constants. Find \(f^{\prime}(N)\), and determine where the predation rate is increasing and where it is decreasing.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=-3 \sin \left(\frac{\pi}{3} x\right)+4 \cos \left(-\frac{\pi}{4} x\right) $$

Show that \(f(x)=-\left|x^{2}-4\right|\) has local maxima at \(x=2\) and \(x=-2\) but \(f(x)\) is not differentiable at \(x=2\) or \(x=-2\).

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=x+\cos x, x \in \mathbf{R} $$

Parasitoids are insects that lay their eggs in, on, or close to other (host) insects. Parasitoid larvae then devour the host insect. The likelihood of escaping parasitism may depend on parasitoid density. One model expressing this dependence sets the probability of escaping parasitism equal to $$ f(P)=e^{-a P} $$ where \(P\) is the parasitoid density and \(a\) is a positive constant. Determine whether the probability of escaping parasitism increases or decreases with parasitoid density.

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

Graph $$ f(x)=|1-| x||, \quad-1 \leq x \leq 2 $$ and determine all local and global extrema on \([-1,2]\).

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=\tan x-x, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $$

As an alternative to the model set forth in Problem 31 , another model sets the probability of escaping parasitism equal to $$ f(P)=\left(1+\frac{a P}{k}\right)^{-k} $$ where \(P\) is the parasitoid density and \(a\) and \(k\) are positive constants. Determine whether the probability of escaping parasitism increases or decreases with parasitoid density.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\sec ^{2}(-4 x) $$

Graph $$ f(x)=-|| x|-2|, \quad-3 \leq x \leq 3 $$ and determine all local and global extrema on \([-3,3]\).

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

Determine whether the functions have absolute maxima and minima, and, if so, find their coordinates. Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. $$ y=\frac{x^{2}-1}{x^{2}+1}, x \in \mathbf{R} $$

Suppose that the height \(y\) in feet of a tree as a function of the age \(x\) in years of the tree is given by $$ y=117 e^{-10 / x}, \quad x>0 $$ (a) Show that the height of the tree increases with age. What is the maximum attainable height? (b) Where is the graph of height versus age concave up, and where is it concave down? (c) Use a graphing calculator to sketch the graph of height versus age. (d) Use a graphing calculator to verify that the rate of growth is greatest at the point where the graph in (c) changes concavity.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=\sec ^{2}\left(\frac{x}{3}\right) $$