Chapter 5

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

Murray's Law for Plants This problem is based on McCulloh et \mathrm{\\{} a l . ~ ( 2 0 0 3 ) . ~ T h e ~ p l a n t ~ x y l e m ~ i s ~ a ~ t r a n s p o r t ~ n e t w o r k ~ w i t h i n ~ p l a n t s ~ that forms a network like the blood vessels of animals. The xylem transports water from the roots, up the plant stem to its leaves. Unlike blood vessels, in some plants the xylem vessels are not single tubes, but are made up of bundles of smaller tubes. Larger xylem vessels contain more tubes, smaller vessels contain fewer tubes. Because vessels are made of smaller tubes, the way that transport costs depend on vessel radius is different for the xylem than for the blood vessels of an animal. Specifically, it can be shown that the cost of transporting water at a flow rate \(f(\) measured in milliliters/s) in a xylem vessel of radius \(r\) and length \(\ell\) (both measured in \(\mathrm{cm}\) ) is given by the function $$ T(r)=0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{2}} $$ where \(r_{T}\) is the radius of one of the tubes within the xylem vessel (you may assume that \(\left.r_{T}=5 \times 10^{-2} \mathrm{~cm}\right)\). (a) Assume that the cost of building the xylem vessel is still proportional to its volume: $$ M(r)=b \pi r^{2} \ell $$ where \(b\) is the metabolic cost of building and maintaining \(1 \mathrm{~cm}^{3}\) of xylem vessel. If the plant controls xylem vessel radius to minimize the total cost \(T(r)+M(r)\), derive a formula relating xylem radius \(r\) to flow rate \(f\). Your formula will include \(b\) as an unknown coefficient. (b) If a xylem vessel of radius \(R\) branches into two smaller vessels of radii \(r_{1}\) and \(r_{2}\), and all vessels minimize the total cost of transport and maintenance, show that the xylem vessel radii are related by Murray's law for plants: $$ R^{2}=r_{1}^{2}+r_{2}^{2} $$

Find the general antiderivative of the given function. $$ f(x)=2 \sin \left(\frac{\pi}{2} x\right)-3 \cos \left(\frac{\pi}{2} x\right) $$

Determine all inflection points. \(f(x)=x \ln x, x>0\)

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 \(x=-2\).

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

Find the general antiderivative of the given function. $$ f(x)=-3 \sin \left(\frac{\pi}{3} x\right)+4 \cos \left(-\frac{\pi}{4} x\right) $$

[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 \(x=0\).

For a population growing according to the logistic model we can calculate a per capita reproductive rate, which is defined to be equal to: $$ g(N)=\frac{f(N)}{N}=r\left(1-\frac{N}{K}\right), \quad N \geq 0 $$ (a) Plot the function \(g(N)\) for \(r=3\) and \(K=10\). (b) For the parameters \(r=3\) and \(K=10\), use calculus to find \(g^{\prime}(N)\), and determine where the function \(g(N)\) is increasing and where it is decreasing. (c) Now suppose that \(K=10\), but the value of \(r\) is not given to you (You may assume \(r>0 .\) ) Show that the reproduction rate \(g(N)\) is a decreasing function of \(N\) for all \(N>0\).

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 \(x=-1\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \begin{array}{l} \lim _{x \rightarrow 0^{+}} x^{\alpha} \ln x, \alpha>0 \text { is any }\\\ \text { positive constant } \end{array} $$

Find the general antiderivative of the given function. $$ f(x)=\sec ^{2}(2 x) $$

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 maximum value.

Growth rates for many microbes and plants depend on the amount of nutrients that are available to them. Monod (1949) introduced a model, now widely adopted, for how the rate of growth of \(E\). coli bacteria depends on the level of glucose in the medium in which the bacteria are grown. Specifically, Monod observed that the reproduction rate of the bacteria (number of cell divisions in one hour) is given as a function of the glucose concentration ( \(C\), measured in units of \(\mathrm{mM}\) ) by an equation: $$ r(C)=1.35 \frac{C}{C+0.022}, \quad C>0 $$ Is \(r(C)\) an increasing function of \(C ?\)

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

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

Find the general antiderivative of the given function. $$ f(x)=\sec ^{2}(4 x) $$

Hill's equation for the oxygen saturation of blood states that the level of oxygen saturation (fraction of hemoglobin molecules that are bound to oxygen) in blood can be represented by a function: $$ f(P)=\frac{P^{n}}{P^{n}+30^{n}} $$ where \(P\) is the oxygen concentration around the blood \((P \geq 0)\) and \(n\) is a parameter that varies between different species. (a) Assume that \(n=1\). Show that \(f(P)\) is an increasing function of \(P\) and that \(f(P) \rightarrow 1\) as \(P \rightarrow \infty\). (b) Assuming that \(n=1\) show that \(f(P)\) has no inflection points. Is it concave up or concave down everywhere? (c) Knowing that \(f(P)\) has no inflection points, could you deduce which way the curve bends (whether it is concave up or concave down) without calculating \(f^{\prime \prime}(P) ?\) (d) For most mammals \(n\) is close to 3. Assuming that \(n=3\) show that \(f(P)\) is an increasing function of \(P\) and that \(f(P) \rightarrow 1\) as \(P \rightarrow \infty\) (e) Assuming that \(n=3\), show that \(f(P)\) has an inflection point, and that it goes from concave up to concave down at this inflection point. (f) Using a graphing calculator plot \(f(P)\) for \(n=1\) and \(n=3\). How do the two curves look different?

Monod Growth Monod's equation describes the rate of growth of microorganisms or plants as a function of the amount, \(C\), of nutrients available to the organisms. The most general form of the equation includes two coefficients, \(a\) and \(K\) : $$ r(C)=\frac{a C}{C+K}, \quad C>0 $$ You may assume that \(a>0\) and \(K>0 .\) Using different values of these coefficients the equation can be used to model different species and different types of nutrients. Show that for any value of \(a\) and any value of \(K\) the reproductive rate \(r(C)\) is an increasing function of \(C\).

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

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \begin{array}{l} \lim _{x \rightarrow+\infty} \frac{\ln x}{x^{\alpha}}, \alpha>0 \text { is any }\\\ \text { positive constant } \end{array} $$

Find the general antiderivative of the given function. $$ f(x)=\tan \left(\frac{x}{3}\right) $$

A two-compartment model of how drugs are absorbed into the body predicts that the amount of drug in the blood will vary with time according to the following function: $$ M(t)=a\left(e^{-k t}-e^{-3 k t}\right), \quad t \geq 0 $$ where \(a>0\) and \(k>0\) are parameters that vary depending on the patient and the type of drug being administered. For parts (a)-(c) of this question you should assume that \(a=1\). (a) Show that \(M(t) \rightarrow 0\) as \(t \rightarrow \infty\) (b) Show that the function \(M(t)\) has a single local maximum, and find the maximum concentration of drug in the patient's blood. (c) Show that the \(M(t)\) has a single inflection point (which you should find). Does the function go from concave up to concave down at this inflection point or vice versa? (d) Would any of your answers to (a)-(c) be changed if \(a\) were not equal to \(1 .\) Which answers?

Suppose the size of a population at time \(t\) is \(N(t)\) and its growth rate is given by the logistic growth model $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right), \quad t \geq 0 $$ where \(r\) and \(K\) are positive constants. (a) Graph the growth rate of the population \(\frac{d N}{d t}\) as a function of population size, \(N\), assuming that \(r=2\) and \(K=100\), and find the population size for which the growth rate is maximal. (b) Show that whatever the value of the parameters \(N\) and \(K\), \(f(N)=r N(1-N / K), N \geq 0\), is differentiable for \(N>0\), and compute \(f^{\prime}(N)\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow+\infty}\left(e^{x}-x^{3}\right) $$

Find the general antiderivative of the given function. $$ f(x)=\tan \left(\frac{x}{4}\right) $$

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 the host insect escaping from being eaten depends on the number of parasitoids in her vicinity. One model for this dependence is that the probability of escaping parasitism is equal to $$ f(P)=e^{-a P} $$ where \(P\) is the number of parasitoids in the host insect's vicinity and \(a\) is a positive constant. Determine whether the probability of the host insect escaping being eaten increases or decreases with the number of parasitoids nearby.

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow+\infty}\left(e^{x}-x^{n}\right), n \in \mathbf{N} $$

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

Suppose that a patient is receiving a particular drug at a constant rate by intravenous line (a needle that delivers the blood directly into one of the patient's veins). In Section \(5.9\) we will show that one model for how the amount of drug in the patient's blood varies with time, \(t\), is: $$ M(t)=a-a e^{-k t}, \quad t>0 $$ This model contains two coefficients; \(a>0\) depends on rate at which the drug is introduced through the intravenous line, and \(k>0\) represents the rate at which it is broken down within the body. Assume that for one particular drug \(a=2\), but the value of \(k\) is not known. (a) Show that whatever the value of \(k\) is, the amount of drug \(M(t)\) is an increasing function of time. (b) Show that whatever the value of \(k\) is, the amount of drug \(M(t)\) increases at a decreasing rate with time, meaning that \(M(t)\) is concave down.

\(f(x)=x^{3}, x \in[0,1]\). (a) Find the slope of the secant line connecting the points \((x, y)=(0,0)\) and \((1,1)\) (b) Find a number \(c \in(0,1)\) 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 \((0,1)\).

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

Find the general antiderivative of the given function. $$ f(x)=\cos ^{2} x-\sin ^{2} x $$

If instead of receiving a drug by intravenous line, a patient takes the drug in pill form then the model from Problem 35 must be modified. The amount of drug in a patient's blood is often modeled by the following equation: $$ M(t)=a e^{-k_{1} t}-a e^{-k_{2} t}, \quad t>0 $$ This model contains three coefficients: \(a>0\) is a measure of the total amount of drug taken, \(k_{1}>0\) is the rate at which the drug is absorbed into the blood from the patient's gut, and \(k_{2}>0\) is the rate at which the drug is broken down by the body. (a) Assuming initially that you know that \(k_{1}=1\) and \(k_{2}=2\), show that there is an interval containing \(t=0\) over which the amount of drug increases with time, whatever the value of \(a\) is. (b) Suppose instead that you know that \(a=1\), and that \(k_{1}=1\) but you do not know \(k_{2}\). However, you do know that \(k_{2}>k_{1}\), meaning that the drug is broken down by the body more rapidly than it is absorbed from the gut. Show that provided \(t<\frac{\ln k_{2}}{k_{2}-1}\), \(M(t)\) is an increasing function of \(t .\) In other words, there is an initial phase after taking the pill where the amount of drug in the patient's blood increases with time. (c) Under the assumptions of part (b) what happens to the amount of drug in the patient's blood if \(t>\frac{\ln k_{2}}{k_{2}-1}\) ?

. Suppose \(f(x)=e^{x}, x \in[0,1]\). (a) Find the slope of the secant line connecting the points \((x, y)=(0,1)\) and \((1, e) .\) (b) Find a number \(c \in(0,1)\) 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 \((0,1)\).

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

Find the general antiderivative of the given function. $$ f(x)=x^{-7}+3 x^{5}+\sin (2 x) $$

Plants employ two basic reproductive strategies: polycarpy, in which reproduction occurs repeatedly during the lifetime of the organism, and monocarpy, in which the plant flowers and produces seeds only once before dying. (Bamboo, for instance, is a monocarpic plant.) Iwasa et al. (1995) argued that the best strategy for a plant depends on how reproductive success (that is, number of progeny that the plant produces) varies with the investment (that is amount of resource that the plant uses up to reproduce) The optimal strategy is polycarpy if reproductive success increases with the investment at a decreasing rate, [or] monocarpy 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.

. Suppose that \(f(x)=x^{2}, x \in[-1,1]\). (a) Find the slope of the secant line connecting the points \((x, y)=(-1,1)\) 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 \((-1,1)\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0^{+}}\left(e^{x}-\frac{1}{x}\right) $$

Find the general antiderivative of the given function. $$ f(x)=2 e^{-3 x}+\sec ^{2}\left(\frac{x}{2}\right) $$

Iwasa et al. (1995) argued that the number of times that a plant can expect to be visited by pollinating insects will depend on the number, \(F\), of flowers that the plant makes. They assumed a power law dependence; namely that the number of pollinator visits is given by: $$ X(F)=c F^{\gamma} $$ where \(c\) and \(\gamma\) are positive constants. (a) Show that if \(\gamma=1 / 2\) then, for all values of \(c\), the average number of pollinator visits to a plant increases with the number of flowers, \(F\), but the rate of increase decreases with \(F\). (b) Show that if \(\gamma=3 / 2\) then, for all values of \(c\), the average number of pollinator visits to a plant increases with the number of flowers, \(F\), and the rate of increase increases with \(F\).

Suppose that \(f(x)=\ln x, x \in[1, e]\). (a) Find the slope of the secant line connecting the points \((x, y)=(1,0)\) and \((e, 1)\) (b) Find a number \(c \in(1, e)\) 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 \((1, e)\).

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+1}\right) $$

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

\mathrm{\\{} L \text { loyd } (1987) studied how the likelihood of a chick surviving to adulthood depends on the amount of resource that the chick's parents invested in it. He proposed the following model for the relationship between the amount of resource \(R\) invested in the chick and the likelihood \(p(R)\) that it survives to adulthood: $$ p(R)=\frac{R^{2}}{k^{2}+R^{2}}, \quad R>0 $$ In the model, \(k>0\) is a coefficient that varies between different species and different environments (a) Show that for all values of \(k, p(R)\) is an increasing function of \(R\) (b) Assume now that \(k=1\). Show that if \(R>1 / \sqrt{3}\), then the \(p(R)\) is concave down. Explain why this means that for \(R\) in this interval, there are diminishing returns from increasing the investment in the chick.

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

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{\sin x}-\frac{1}{x}\right) $$

Find the general antiderivative of the given function. $$ f(x)=5 e^{3 x}-\sec ^{2}(x-3) $$

A pulse-chase experiment can be used to see how a particular chemical is processed by the cell; for example, how pancreatic cells convert amino acids into insulin. The experiment starts with a pulse phase, in which the cells are fed a radioactively labelled form of the amino acid. Following this there is a chase phase, in which they are fed the same amino acid, but without the radioactive label. Any insulin that the cells produce is removed and tested for the presence of the radioactive label. You perform this experiment for pancreatic cells that have been treated with different drugs before the pulse-chase experiment. You measure the amount of radioactive labelled insulin \(c(t)\) produced as a function of time, \(t\) : $$ c(t)=12.7 t e^{-k t}, \quad t>0 $$ where \(k>0\) is a coefficient that depends on which drugs the cells have been treated with before the pulse chase experiment. You expect the level of radioactive label in the insulin to start increasing (during the pulse phase), and then decrease (during the chase phase), as the radioactive amino acid works its way through the cell and is replaced by the non-radioactive amino acid. Show that the function \(c(t)\) has this behavior for all values of the coefficient \(k\).