Chapter 5

Suppose the size of a population at time \(t\) is \(N(t)\) and its growth rate 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 \(r=2\) and \(K=100\), and find the population size for which the growth rate is maximal. (b) Show that \(f(N)=r N(1-N / K), N \geq 0\), is differentiable for \(N>0\), and compute \(f^{\prime}(N)\). (c) Show that \(f^{\prime}(N)=0\) for the value of \(N\) that you determined in (a) when \(r=2\) and \(K=100\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 1^{-}}(1-x) \tan \left(\frac{\pi}{2} x\right) $$

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. $$ y=\ln \left(x^{2}+1\right), x \in \mathbf{R} $$

Plants employ two basic reproductive strategies: polycarpy, in which reproduction occurs repeatedly during the lifetime of the organism, and monocarpy, in which reproduction occurs only once during the lifetime of the organism. (Bamboo, for instance, is a monocarpic plant.) The following quote is taken from Iwasa et al. (1995): The optimal strategy is polycarpy (repeated reproduction) if reproductive success increases with the investment at a decreasing rate, [or] monocarpy ("big bang" reproduction) or intermittent reproduction if the reproductive success increases at an increasing rate. (a) Sketch the graph of reproductive success as a function of reproductive investment for the cases of (i) polycarpy and (ii) monocarpy. (b) Given that the second derivative describes whether a curve bends upward or downward, explain the preceding quote in terms of the second derivative of the reproductive success function.

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

Suppose that the size of a population at time \(t\) is \(N(t)\) and its growth rate 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 $$ g(N)=\frac{1}{N} \frac{d N}{d t} $$ (a) Show that $$ g(N)=r\left(1-\frac{N}{K}\right) $$ (b) Graph \(g(N)\) as a function of \(N\) for \(N \geq 0\) when \(r=2\) and \(K=100\), and find the population size for which the per capita growth rate is maximal.

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 \(i t\) 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.

Assume that the formula (Iwasa et al., 1995 ) $$ X(F)=c F^{\gamma} $$ where \(c\) is a positive constant, expresses the relationship between the number of flowers on a plant, \(F\), and the average number of pollinator visits, \(X(F)\). Find the range of values for the parameter \(\gamma\) such that the average number of pollinator visits to a plant increases with the number of flowers \(F\) but the rate of increase decreases with \(F\). Explain your answer in terms of appropriate derivatives of the function \(X(F)\).

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

Suppose \(f(x)=x^{2}, x \in[0,2]\). (a) Find the slope of the secant line connecting the points \((0,0)\) and \((2,4)\). (b) Find a number \(c \in(0,2)\) such that \(f^{\prime}(c)\) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in \((0,2)\).

Let $$ f(x)=-\frac{2}{x^{2}-1}, \quad x \neq-1,1 $$ (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 \(f(x)\)

Assume that the dependence of the averag number of pollinator visits to a plant, \(X\), on the number of flowers \(F\), is given by $$ X(F)=c F^{\gamma} $$ where \(\gamma\) is a positive constant less than 1 and \(c\) is a positive constant (Iwasa et al., 1995 ). How does the average number o pollen grains exported per flower, \(E(F)\), change with the numbe of flowers on the plant, \(F\), if \(E(F)\) is proportional to $$ 1-\exp \left[-k \frac{X(F)}{F}\right] $$ where \(k\) is a positive constant?

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

Suppose \(f(x)=1 / x, x \in[1,2]\). (a) Find the slope of the secant line connecting the points \((1,1)\) and \((2,1 / 2)\). (b) Find a number \(c \in(1,2)\) such that \(f^{\prime}(c)\) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in \((1,2)\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}}(\cot x-\csc x) $$

Let $$ f(x)=\frac{2 x^{2}-5}{x+2}, \quad x \neq-2 $$ (a) Show that \(x=-2\) is a vertical asymptote. (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) Since the degree of the numerator is one higher than the degree of the denominator, \(f(x)\) has an oblique asymptote. Find it. (e) Sketch the graph of \(f(x)\) together with its asymptotes.

Denote the size of a population by \(N(t)\), and assume that \(N(t)\) satisfies $$ \frac{d N}{d t}=N e^{-a N}-N^{2} $$ where \(a\) is a positive constant. (a) Show that the nontrivial equilibrium \(N^{*}\) satisfies $$ e^{-a N^{*}}=N^{*} $$ (b) Assume now that the nontrivial equilibrium \(N^{*}\) is a function of the parameter \(a\). Use implicit differentiation to show that \(N^{*}\) is a decreasing function of \(a\).

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x^{-7}+3 x^{5}+\sin (2 x) $$

Suppose that \(f(x)=x^{2}, x \in[-1,1]\). (a) Find the slope of the secant line connecting the points \((-1,1)\) and \((1,1)\). (b) Find a number \(c \in(-1,1)\) such that \(f^{\prime}(c)\) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in \((-1,1)\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-1}\right) $$

Let $$ f(x)=\frac{\sin x}{x}, \quad x \neq 0 $$ (a) Show that \(y=0\) is a horizontal asymptote. (b) Since \(f(x)\) is not defined at \(x=0\), does this mean that \(f(x)\) has a vertical asymptote at \(x=0\) ? Find \(\lim _{x \rightarrow 0^{+}} f(x)\) and \(\lim _{x \rightarrow 0^{-}} f(x) .\) (c) Use a graphing calculator to sketch the graph of \(f(x)\).

Denote the size of a population by \(N(t)\), and assume that \(N(t)\) satisfies $$ \frac{d N}{d t}=N\left(1-\frac{N}{K}\right)-N \ln N $$ where \(K\) is a positive constant. (a) Show that if \(K>1\), then there exists a nontrivial equilibrium \(N^{*}>0\) that satisfies $$ 1-\frac{N^{*}}{K}=\ln N^{*} $$ (b) Assume now that the nontrivial equilibrium \(N^{*}\) is a function of the parameter \(K\). Use implicit differentiation to show that \(N^{*}\) is an increasing function of \(K\).

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

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

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).

(Adapted from Bellows, 1981 ) Suppose that a study plot contains \(N\) annual plants, each of which produces \(S\) seeds that are sown within the same plot. The number of surviving plants in the next year is given by $$ A(N)=\frac{N S}{1+(a N)^{b}} $$ for some positive constants \(a\) and \(b .\) This mathematical model incorporates density-dependent mortality: The greater the number of plants in the plot, the lower is the number of surviving offspring per plant, which is given by \(A(N) / N\) and is called the net reproductive rate. (a) Use calculus to show that \(A(N) / N\) is a decreasing function of \(N\). (b) The following quantity, called the \(k\) -value, can be used to quantify the effects of intraspecific competition (i.e., competition between individuals of the same species): $$ k=\log [\text { initial density }]-\log [\text { final density }] $$ Here, "log" denotes the logarithm to base \(10 .\) The initial density is the product of the number of plants \((N)\) and the number of seeds each plant produces \((S)\). The final density is given by \((5.6)\). Use the expression for \(k\) and \((5.6)\) to show that $$ \begin{aligned} k &=\log [N S]-\log \left[\frac{N S}{1+(a N)^{b}}\right] \\ &=\log \left[1+(a N)^{b}\right] \end{aligned} $$ We typically plot \(k\) versus \(\log N ;\) the slope of the resulting curve is then used to quantify the effects of competition. (i) Show that $$ \frac{d \log N}{d N}=\frac{1}{N \ln 10} $$ where \(\ln\) denotes the natural logarithm. (ii) Show that $$ \frac{d k}{d \log N}=(\ln 10) N \frac{d k}{d N}=\frac{b}{1+(a N)^{-b}} $$ (iii) Find $$ \lim _{N \rightarrow \infty} \frac{d k}{d \log N} $$ (iv) Show that if $$ \frac{d k}{d \log N}<1 $$ then \(A(N)\) is increasing, whereas if $$ \frac{d k}{d \log N}>1 $$ then \(A(N)\) is decreasing. [Hint: Compute \(A^{\prime}(N) .\) ] Explain in words what the two inequalities mean with respect to varying the initial density of seeds and observing the number of surviving plants the next year. (Hint: The first case is called undercompensation and the second case is called overcompensation.) (v) The case $$ \frac{d k}{d \log N}=1 $$ is referred to as exact compensation. Suppose that you plot \(k\) versus \(\log N\) and observe that, over a certain range of values of \(N\), the slope of the resulting curve is equal to \(1 .\) Explain what this means.

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

Let \(f(x)=x(1-x)\). Use the MVT to find an interval that contains a number \(c\) such that \(f^{\prime}(c)=0\).

Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{\sin ^{2} x}-\frac{1}{x}\right) $$

Let $$ f(x)=\frac{x^{k}}{1+x^{k}}, x \geq 0 $$ where \(k\) is a positive integer greater than \(1 .\) (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 \(\lim _{x \rightarrow \infty} f(x)\) 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).

(Adapted from Reiss, 1989 ) Suppose that the rate at which body weight \(W\) changes with age \(x\) is $$ \frac{d W}{d x} \propto W^{a} $$ where \(a\) is some species-specific positive constant. (a) The relative growth rate (percentage weight gained per unit of time ) is defined as $$ \frac{1}{W} \frac{d W}{d x} $$ What is the relationship between the relative growth rate and body weight? For which values of \(a\) is the relative growth rate increasing, and for which values is it decreasing? (b) As fish grow larger, their weight increases each day but the relative growth rate decreases. If the rate of growth is described by (5.7), what values of \(a\) can you exclude on the basis of your results in (a)? Explain how the increase in percentage weight (relative to the current body weight) differs for juvenile fish and for adult fish.

In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=5 e^{3 x}-\sec ^{2}(x-3) $$

Let \(f(x)=1 /\left(1+x^{2}\right)\). Use the MVT to find an interval that contains a number \(c\) such that \(f^{\prime}(c)=0\).

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

Let $$ f(x)=\frac{x}{a+x}, x \geq 0 $$ where \(a\) is a positive constant. (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 \(\lim _{x \rightarrow \infty} f(x)\) 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).

Allometric Growth Allometric equations describe the scaling relationship between two measurements, such as tree height versus tree diameter or skull length versus backbone length. These equations are often of the form $$ Y=b X^{a} $$ where \(b\) is some positive constant and \(a\) is a constant that can be positive, negative, or zero. (a) Assume that \(X\) and \(Y\) are body measurements (and therefore positive) and that their relationship is described by an allometric equation of the form (5.8). For what values of \(a\) is \(Y\) an increasing function of \(X\), but one such that the ratio \(Y / X\) decreases with increasing \(X ?\) Is \(Y\) concave up or concave down in this case? (b) In vertebrates, we typically find \([\) skull length \(] \propto[\text { body length }]^{a}\) for some \(a \in(0,1) .\) Use your answer in (a) to explain what this means for skull length versus body length in juveniles versus adults; that is, at which developmental stage do vertebrates have larger skulls relative to their body length?

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{(a+1) x}}{a} $$

Suppose that \(f(x)=-x^{2}+2 .\) Explain why there exists a point \(c\) in the interval \((-1,2)\) such that \(f^{\prime}(c)=-1\).

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

Let $$ f(x)=\frac{2}{1+e^{-x}}, x \in \mathbf{R} $$ (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 \(\lim _{x \rightarrow \infty} f(x)\) and decide whether \(f(x)\) has a horizontal asymptote. (d) Find \(\lim _{x \rightarrow-\infty} f(x)\) and decide whether \(f(x)\) has a horizontal asymptote. (e) Sketch the graph of \(f(x)\) together with its asymptotes and inflection points (if they exist).

The pH value of a solution measures the concentration of hydrogen ions, denoted by \(\left[\mathrm{H}^{+}\right]\), and is defined as $$ \mathrm{pH}=-\log \left[\mathrm{H}^{+}\right] $$ Use calculus to decide whether the pH value of a solution increases or decreases as the concentration of \(\mathrm{H}^{+}\) increases.

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\sin ^{2}\left(a^{2} x+1\right) $$

Suppose that \(f(x)=x^{3} .\) Explain why there exists a point \(c\) in the interval \((-1,1)\) such that \(f^{\prime}(c)=1\).

Population Growth Suppose that the growth rate of a population is given by $$ f(N)=N\left(1-\left(\frac{N}{K}\right)^{\theta}\right) N \geq 0 $$ 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 the population size for which the growth rate is maximal.

The differential equation $$ \frac{d y}{d x}=k \frac{y}{x} $$ describes allometric growth, where \(k\) is a positive constant. Assume that \(x\) and \(y\) are both positive variables and that \(y=f(x)\) is twice differentiable. Use implicit differentiation to determine for which values of \(k\) the function \(y=f(x)\) is concave up.

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{1}{a x+3} $$

Sketch the graph of a function \(f(x)\) that is continuous on the closed interval \([0,1]\) and differentiable on the open interval \((0,1)\) such that there exists exactly one point \((c, f(c))\) on the graph at which the slope of the tangent line is equal to the slope of the secant line connecting the points \((0, f(0))\) and \((1, f(1))\). Why can you be sure that there is such a point?

Let \(N(t)\) denote the population size at time \(t\), and assume that \(N(t)\) is twice differentiable and satisfies the differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a real number. Differentiate the differential equation with respect to \(t\), and state whether \(N(t)\) is concave up or down.

In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{a}{a+x} $$

Sketch the graph of a function \(f(x)\) that is continuous on the closed interval \([0,1]\) and differentiable on the open interval \((0,1)\) such that there exist exactly two points \(\left(c_{1}, f\left(c_{1}\right)\right)\) and \(\left(c_{2}, f\left(c_{2}\right)\right)\) on the graph at which the slope of the tangent lines is equal to the slope of the secant line connecting the points \((0, f(0))\) and \((1, f(1)) .\) Why can you be sure that there is at least one such point?