Chapter 5

Initially you measure that a colony of bacterial cells contains 2000 cells. 2 hours later you measure the colony again, and count 4000 cells. (a) How many cells would you expect the colony to contain 3 hours after the start of the experiment? (b) In fact, you realize that the hemocytometer that you used to count the cells for both measurements is only accurate to \(10 \%\), meaning that if you count 1000 cells, the real number of cells is somewhere between \(1000-100=900\) cells and \(1000+100=\) 1100 cells. What is the largest possible number of cells in the colony 3 hours after the start of the experiment? And what is the smallest possible number of cells at 3 hours?

Use the Newton-Raphson method to find a numerical approximation for all of the solutions of: $$ x^{4}+x^{3}+1=x^{2}+2 x $$ correct to six decimal places.

Find the equilibria of $$x_{t+1}=\frac{1}{6}\left(x_{t}^{2}+x_{t}+4\right), \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether they are stable or unstable.

Find the general antiderivative of the given function. $$ f(x)=4 x^{3}-2 x+3 $$

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\sqrt{2 x+1}, x \geq-1 / 2 $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extreme- value theorem. With the help of a graphing calculator or spreadsheet, graph each function and locate its global extrema. (Note that a function may have more than one global minimum or maximum point.) $$ f(x)=e^{-|x|},-1 \leq x \leq 1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} x^{2} \ln x $$

Suppose that \(a\) and \(b\) are the side lengths in a right triangle whose hypotenuse is \(10 \mathrm{~cm}\) long. Show that the area of the triangle is largest when \(a=b\).

Use the Newton-Raphson method to solve the equation $$ \sin x=\frac{1}{2} x $$ in the interval \((0, \pi)\).

Find the equilibria of $$x_{t+1}=\sqrt{x_{t}+1}+1, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether they are stable or unstable (you may assume that \(\left.x_{t} \geq-1\right)\)

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

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=(3 x-1)^{1 / 3}, x \in \mathbf{R} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ , each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extreme- value theorem. With the help of a graphing calculator or spreadsheet, graph each function and locate its global extrema. (Note that a function may have more than one global minimum or maximum point.) $$ f(x)=x \ln x, 1 \leq x \leq 2 $$

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

A rectangle has its base on the \(x\) -axis, its lower left corner at \((0,0)\), and its upper right corner on the curve \(y=1 / x\). What is the smallest perimeter the rectangle can have?

Concentration of Adderall in Blood The drug Adderall (a proprietary combination of amphetamine salts) is used to treat ADHD. Adderall has first order kinetics for elimination, with elimination rate constant \(k_{1}=0.08 \mathrm{hr}\). We will assume that pills are quickly absorbed into the blood; that is, when the patient takes a pill their blood concentration of the drug immediately jumps. (a) Assuming that the patient takes one pill at \(8 \mathrm{am}\), the concentration in their blood after taking the pill is \(33.8 \mathrm{ng} / \mathrm{ml}\). Assuming that they take no other pills during the day, write down and then solve the differential equation that gives the concentration of drug in the blood, \(M(t)\), over the course of a day. (Hint: You may find it helpful to define time \(t\) by the number of hours elapsed since \(8 \mathrm{am} .)\) (b) What is the blood concentration just before the patient takes their next dose of the drug, at \(8 \mathrm{am}\) the next day? (c) At what time during the day does the blood concentration fall to half of its initial value? (d) In an alternative treatment regimen, the patient takes two half pills, one every 12 hours. They take the first pill at \(8 \mathrm{am}\), with no drug in their system. Why would we expect their drug concentration to be \(16.9 \mathrm{ng} / \mathrm{ml}\) immediately after taking the pill? (e) What is the concentration in their blood at \(8 \mathrm{pm} ?\) (f) At \(8 \mathrm{pm}\), the patient takes the other half pill. This increases the concentration of Adderall in their blood by \(16.9 \mathrm{ng} / \mathrm{ml}\) (i.e.. the concentration increases at \(8 \mathrm{pm}\) by \(16.9 \mathrm{ng} / \mathrm{ml}\) ). Derive a formula for the concentration in their blood as a function of time elapsed since \(8 \mathrm{am}\). (You will need different expressions for the concentration for \(0

Find the equilibria of $$x_{t+1}=4 x_{t}^{2}+x_{t}-1, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether they are stable or unstable.

Find the local maxima and minima of each of the functions. Determine whether each function has local 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. \(y=e^{-x^{2}},-1 \leq x \leq 1\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{1}{x}, x \neq 0 $$

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

Denote by \((x, y)\) a point on the straight line \(y=4-3 x .\) (See Figure \(5.55 .\) ) (a) Show that the distance from \((x, y)\) to the origin is given by $$ f(x)=\sqrt{x^{2}+(4-3 x)^{2}} $$ (b) Give the coordinates of the point on the line \(y=4-3 x\) that is closest to the origin. (Hint: Find \(x\) so that the distance you computed in (a) is minimized.) (c) Show that the square of the distance between the point \((x, y)\) on the line and the origin is given by $$ g(x)=[f(x)]^{2}=x^{2}+(4-3 x)^{2} $$ and find the minimum of \(g(x)\). Show that this minimum agrees with your answer in (b).

In Problem 9 we neglected to consider the time delay between a pill being taken and the drug entering the patient's blood. In Chapter 8 we will introduce compartment models as models for drug absorption. We will show that a good model for a drug being absorbed from the gut is that the rate of drug absorption, \(A(t)\), varies with time according to: $$ A(t)=C e^{-k t}, t \geq 0 $$ where \(C>0\) and \(k>0\) are coefficients that will depend on the type of drug, as well as varying between patients. (a) Assume that the drug has first order elimination kinetics, with elimination rate \(k_{1} .\) Show that the amount of drug in the patient's blood will obey a differential equation: $$ \frac{d M}{d t}=C e^{-k t}-k_{1} M $$ (b) Verify that a solution of this differential equation is: $$ M(t)=\frac{C e^{-k t}}{k_{1}-k}+a e^{-k_{1} t} $$ where \(a\) is any coefficient, and we assume \(k_{1} \neq k\). (c) To determine the coefficient \(a\), we need to apply an initial condition. Assume that there was no drug present in the patient's blood when the pill first entered the gut (that is, \(M(0)=0\) ). Find the value of \(a\). (d) Let's assume some specific parameter values. Let \(C=2\), \(k=3\), and \(k_{1}=1 .\) Show that \(M(t)\) is initially increasing, and then starts to decrease. Find the maximum level of drug in the patient's blood. (e) Show that \(M(t) \rightarrow 0\) as \(t \rightarrow \infty\). (f) Using the information from (d) and (e), make a sketch of \(M(t)\) as a function of \(t\).

Find the equilibrium point of $$x_{t+1}=x_{t}+\frac{1}{2} e^{-x_{t}}-\frac{1}{2}, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether it is stable or unstable.

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

Find the local maxima and minima of each of the functions. Determine whether each function has local 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. \(y=\cos \left(\pi x^{2}\right),-1 \leq x \leq 1\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{x+1}{x}, x \neq 0 $$

10\. Sketch the graph of a function that is continuous on the closed interval \([-2,1]\) and has a global maximum and a global minimum in the interior of the interval.

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

How close does the line \(y=1+2 x\) come to the origin?

Find the equilibria of $$x_{t+1}=\frac{2 x_{t}}{1+x_{t}}, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibríum point to determine whether they are stable or unstable.

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

Find the local maxima and minima of each of the functions. Determine whether each function has local 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. \(y=e^{-|x|}, x \in \mathbf{R}\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ \left(x^{2}+1\right)^{1 / 3}, x \in \mathbf{R} $$

Sketch the graph of a function that is continuous on the open interval \((0,1)\) and has a global maximum but does not have a global minimum.

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

How close does the curve \(y=1 / x\) come to the origin? (Hint: Find the point on the curve that minimizes the square of the distance between the origin and the point on the curve. If you use the square of the distance instead of the distance, you avoid dealing with square roots.)

Find the equilibria of $$x_{t+1}=\frac{x_{t}}{0.3+x_{t}}, \quad t=0,1,2, \ldots$$ and use the stability criterion for an equilibrium point to determine whether they are stable or unstable.

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

Find the local maxima and minima of each of the functions. Determine whether each function has local 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. \(y=\sin 2 \pi x, 0 \leq x \leq 1\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{5}{x-2}, x \neq 2 $$

Sketch the graph of a function that is continuous on the closed interval \([0,4]\), except at \(x=2\), and has neither a global maximum nor a global minimum in its domain.

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

How close does the curve \(y=1 / x^{2}\) come to the origin? (Hint: Find the point on the curve that minimizes the square of the distance between the origin and the point on the curve. If you use the square of the distance instead of the distance, you avoid dealing with square roots.)

Ibuprofen in Blood You are modeling the concentration of the drug ibuprofen (Advil) in a person's blood after they take one pill. We assume that after they take the pill the drug enters their blood effectively instantaneously. Ibuprofen has first order elimination kinetics. (a) Explain why the concentration of drug in their blood satisfies a differential equation: $$ \frac{d c}{d t}=-k_{1} c \quad \text { with } \quad c(0)=c_{0} $$ and explain what the constants \(k_{1}\) and \(c_{0}\) represent. (You do not need to solve the differential equation.) (b) You measure the following data for the concentration of ibuprofen in a patient's blood $$ \begin{array}{cc} \hline t \text { (hrs) } & c(t)(\mathrm{mg} / \text { liter }) \\ \hline 0 & 40 \\ 1 & 30.3 \\ \hline \end{array} $$ Write down the solution to the differential equation from part (a). Then calculate the parameters \(c_{0}\) and \(k_{1}\) that fit the model to this data.

(a) Use the stability criterion to characterize the stability of the equilibria of $$x_{t+1}=\frac{5 x_{t}^{2}}{4+x_{t}^{2}}, \quad t=0,1,2, \ldots$$ (b) Use cobwebbing to find the limit that \(x_{t}\) converges to as \(t \rightarrow \infty\) if (i) \(x_{0}=0.5\) and (ii) \(x_{0}=2\).

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

Find the local maxima and minima of each of the functions. Determine whether each function has local 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. \(y=\frac{1}{3} x^{3}+\frac{1}{2} x^{2}+2, x \in \mathbf{R}\)

Use the first derivative test and the second derivative test to determine where each function is increasing, decreasing, concave up, and concave down. You do not need to use a graphing calculator for these exercises. $$ y=\frac{1}{(1+x)^{2}}, x \neq-1 $$

T In Problems 13-18, use a graphing calculator or spreadsheet to plot the function and determine all local and global extrema. $$ f(x)=4-x, x \in[-1,4) $$

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