Chapter 5

Suppose that \(f(x)=e^{-|x|}, x \in[-2,2]\). (a) Show that \(f(-2)=f(2)\). (b) Compute \(f^{\prime}(x)\), where defined. (c) Show that there is no number \(c \in(-2,2)\) such that \(f^{\prime}(c)=0\). (d) Explain why your results in (a) and (c) do not contradict Rolle's theorem. (e) Use a graphing calculator to sketch the graph of \(f(x)\).

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

Find the general solution of the differential equation. $$ \frac{d y}{d s}=\cos (2 \pi s), 0 \leq s \leq 1 $$

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{1}{1-x}, x>1 $$

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

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{1}{x+1}, x>-1 $$

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

Solve the initial-value problem. $$ \frac{d y}{d x}=3 x^{2}, \text { for } x \geq 0 \text { with } y(0)=1 $$

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

Solve the initial-value problem. $$ \frac{d y}{d x}=\frac{x^{2}}{3}, \text { for } x \geq 0 \text { with } y(0)=2 $$

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

Solve the initial-value problem. $$ \frac{d y}{d x}=\sqrt{x}, \text { for } x \geq 0 \text { with } y(1)=2 $$

Use l'Hôpital's rule to find $$ \lim _{x \rightarrow 0} \frac{a^{x}-1}{b^{x}-1} $$ where \(a, b>0\).

Solve the initial-value problem. $$ \frac{d y}{d x}=\frac{2}{\sqrt{x}}, \text { for } x \geq 1 \text { with } y(4)=3 $$

Use l'Hôpital's rule to find $$ \lim _{x \rightarrow \infty}\left(1+\frac{c}{x}\right)^{x} $$ where \(c\) is a constant.

Solve the initial-value problem. $$ \frac{d N}{d t}=\frac{1}{t}, \text { for } t \geq 1 \text { with } N(1)=10 $$

For \(p>0\), determine the values of \(p\) for which the following limit is either 1 or \(\infty\) or a constant that is neither 1 nor \(\infty\) : $$ \lim _{x \rightarrow \infty}\left(1+\frac{1}{x^{p}}\right)^{x} $$

Solve the initial-value problem. $$ \frac{d N}{d t}=\frac{t+2}{t}, \text { for } t \geq 1 \text { with } N(1)=2 $$

Cell Division Time The Gamma distribution is used as a model for the amount of time taken for a cell to undergo a certain number of divisions. According to the Gamma distribution the likelihood that it takes \(t\) hours for the cell to complete all of its divisions is proportional to: $$ f(t)=t^{p} e^{-t} $$ where \(p>-1\) is a constant that depends on the number of cells that are dividing, and on the conditions that the cells are growing in. (a) Show that for all values of \(p, f(t) \rightarrow 0\) as \(t \rightarrow \infty\). (b) For what values of \(p\) does \(f(t)\) converge to a finite value as \(t \rightarrow 0 ?\)

Solve the initial-value problem. $$ \frac{d W}{d t}=e^{t}, \text { for } t \geq 0 \text { with } W(0)=1 $$

Lifespan Modeling The Weibull distribution is used to model the lifespan of organisms. According to the Weibull distribution, the likelihood that an animal dies at the age of \(t\) is proportional to: $$ f(t)=t^{k-1} \exp \left(-t^{k}\right), t \geq 0 $$ where \(k>0\) is a constant that depends on the type of organism being studied, and on the environment that it is living in. (a) Show that for all values of \(k, f(t) \rightarrow 0\) as \(t \rightarrow \infty\). (b) For what values of \(k\) does \(f(t)\) converge to a finite value as \(t \rightarrow 0 ?\)

Solve the initial-value problem. $$ \frac{d W}{d t}=e^{-3 t}, \text { for } t \geq 0 \text { with } W(0)=2 $$

Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0 $$ for any number \(p>0\). This shows that the logarithmic function grows more slowly than any positive power of \(x\) as \(x \rightarrow \infty\).

Solve the initial-value problem. $$ \frac{d W}{d t}=\exp (t+1), \text { for } t \geq 0 \text { with } W(0)=2 / 3 $$

Species Diversity Chapter 3 introduced the Shannon diversity index for the diversity of a habitat. If a population contains two different species of animals, and they are present in the proportions \(p\) and \(1-p\) respectively, then the Shannon diversity index for the population is given by the formula: $$ H(p)=-p \ln p-(1-p) \ln (1-p), \quad 0

Solve the initial-value problem. $$ \frac{d W}{d t}=e^{-5 t}, \text { for } t \geq 0 \text { with } W(0)=1 $$

The height \(y\) in feet of a tree as a function of the tree's age \(x\) in years is given by $$ y=121 e^{-17 / x} \quad \text { for } x>0 $$ (a) Determine (1) the rate of growth when \(x \rightarrow 0^{+}\) and \((2)\) the limit of the height as \(x \rightarrow \infty\). (b) Find the age at which the growth rate is maximal. (c) Show that the height of the tree is an increasing function of age. At what age is the height increasing at an accelerating rate and at what age at a decelerating rate?

Solve the initial-value problem. $$ \frac{d T}{d t}=\sin (\pi t), \text { for } t \geq 0 \text { with } T(0)=3 $$

Solve the initial-value problem. $$ \frac{d T}{d t}=\cos (\pi t), \text { for } t \geq 0 \text { with } T(0)=3 $$

Solve the initial-value problem. $$ \frac{d y}{d x}=\frac{e^{-x}+e^{x}}{2}, \text { for } x \geq 0 \text { with } y=0 \text { when } x=0 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=\frac{1}{y}, \text { for } x \geq 1 \text { with } y(1)=1 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=1-y, \text { for } x \geq 0 \text { with } y(0)=0 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=e^{y}, \text { for } x \geq 0 \text { with } y(0)=0 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=\frac{1}{1-y}, \text { for } x \geq 0 \text { with } y(0)=0 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=\frac{y}{y^{2}+1}, \text { for } x \geq 0 \text { with } y(0)=1 $$

Solve by rewriting the differential equation as an equation for \(\frac{d x}{d y}\) : $$ \frac{d y}{d x}=\frac{y}{y+1}, \text { for } x \geq 0 \text { with } y(0)=1 $$

Suppose that the length of a certain organism at age \(t\) is give by \(L(t)\), which satisfies the differential equation $$ \frac{d L}{d t}=e^{-0.1 t}, \quad t \geq 0 $$ Find \(L(t)\) if the limiting length \(L_{\infty}\) is given by $$ L_{\infty}=\lim _{t \rightarrow \infty} L(t)=25 $$ How big is the organism at age \(t=0\) ?

Fish are indeterminate growers; that is, their length \(L(t)\) increases with age \(t\) throughout their lifetime. If we plot the growth rate \(d L / d t\) versus age \(t\) on semilog paper, a straight line with negative slope results, meaning that: $$ \frac{d L}{d t}=A e^{-k t} $$ where \(A>0\) and \(k>0\) are both coefficients that depend on the species of fish, and the habitat that it is growing in. (a) Find the solution for this differential equation (your solution will include \(A\) and \(k\) as unknown constants, as well as one additional unknown constant \(C\) from the antiderivative). (b) Find the values for the constants \(A, k, C\), that would fit the solution to the following data \(L(0)=5, L(1)=10\), and $$ \lim _{t \rightarrow \infty} L(t)=30. $$ (c) Graph the solution \(L(t)\) as a function of \(t\).

Elimination of ethanol from the blood is known to have zeroth order kinetics. Provided no more ethanol enters the blood, the concentration of ethanol in a person's blood will therefore obey the following differential equation: $$ \frac{d M}{d t}=-k_{0} $$ where for a typical adult \(k_{0}=0.186 \mathrm{~g} /\) liter \(/ \mathrm{hr}\) (al-Lanqawi et al. 1992). (a) Explain why \(M(t)\) can only obey the above differential equation if \(M>0\) (once \(M\) drops to 0 , it is usual to assume that \(\left.\frac{d M}{d t}=0\right)\) (b) If a person's blood alcohol concentration is \(1.6 \mathrm{~g}\) /liter at midnight, what will their blood alcohol concentration be at \(2 \mathrm{am}\) ? You may assume that she drinks no more alcohol after midnight. (c) At what time will their blood alcohol concentration drop to \(0 \mathrm{~g} /\) liter?

Some microbes regulate their growth according to a circadian clock. This clock means that their growth rate fluctuates predictably over the course one day. For example the filamentous fungus Neurospora crassa grows almost twice as fast at night-time than during the day. Gooch, Freeman and \mathrm{\\{} L a k i n - T h o m a s ~ ( 2 0 0 4 ) ~ m e a s u r e d ~ t h e ~ c h a n g e ~ o f ~ g r o w t h ~ r a t e ~ o v e r ~ time. If growth rate is measured in \(\mathrm{mm} /\) day then their data can be fit by the following relationship: $$ \frac{d L}{d t}=38.4+2.4 \cos (4 \pi t)-12 \sin (2 \pi t) $$ where \(L(t)\) is the total size of the fungus, measured in \(\mathrm{mm}\), and \(t\) is the time measured in hours. Calculate: (a) the total extra length added to the fungus between \(t=0\) and \(t=1\) hours. (b) the total extra length added to the fungus between \(t=0\) and \(t=12\) hours. (c) the total extra length added to the fungus between \(t=0\) and \(t=24\) hours.

Suppose that a drug is eliminated so slowly from the blood that its elimination kinetics can be essentially ignored. Then according to Section \(5.9\) the total amount of drug in the blood is given by a differential equation: $$ \frac{d M}{d t}=A(t) $$ where \(A(t)\) is the rate of absorption. We will show in Chapter 8 that if the drug is absorbed into the blood from a pill in the patient's gut, then \(A(t)\) is given by a function $$ A(t)=C e^{-k t} $$ where \(C>0\) and \(k>0\) are constants that depend on the type of the drug being administered. Assume that at \(t=0\) there is no drug present in the patient's blood (i.e., \(M(0)=0\) ). Solve this initial value problem, and, using the methods from Section \(5.6\), sketch the graph of \(M(t)\) against \(t\).