Chapter 5

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ x^{2}-7=0 $$ that is correct to six decimal places.

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

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. \(y=(x-1)^{2},-2 \leq x \leq 3\)

In Problems 1-20, 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. $$ y=2 x-x^{2}, x \in \mathbf{R} $$

\mathrm{\\{} I n ~ P r o b l e m s ~ \(1-8\), 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.) $$ f(x)=2 x, 0 \leq x \leq 1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 5} \frac{x^{2}-25}{x-5} $$

Find the smallest perimeter possible for a rectangle whose area is \(25 \mathrm{in} .^{2}\).

A population of bacteria grows exponentially and has mean time between divisions \(t_{b}=2\) hours. Assume that cell death can be ignored (that is, \(m=0\) ). (a) Sketch on the same axes (of \(N(t)\) against \(t\) ) the size of the population over the interval \(0

The equation $$ x^{2}-5=0 $$ has two solutions. Use the Newton-Raphson method to approximate the two solutions to four decimal places.

(a) Find all equilibria of $$ N_{t+1}=0.7 N_{t}, \quad t=0,1,2, \ldots $$ (b) Use cobwebbing to determine the stability of the equilibria you found in (a).

Find the general antiderivative of the given function. $$ f(x)=5\left(1-x^{2}\right) $$

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. \(y=\sqrt{x+1}, 1 \leq x \leq 2\)

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. $$ y=x^{2}+5 x, x \in \mathbf{R} $$

\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.) $$ f(x)=-x^{2}+1,-1 \leq x \leq 1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$

Show that, among all rectangles with a given perimeter, the square has the largest area.

A population of cells initially contains 1000 cells. Two hours later the population contains 3000 cells. (a) Estimate the division time \(t_{b}\) for this population (you can assume that mortality may be neglected; that is, \(m=0\) ). (b) At what time would we expect the size of the population to reach 6000 cells? (c) If we did not neglect cell death (that is, \(m \neq 0\) ), would our estimate for the division time \(t_{b}\) increase or decrease from the value given in (a)? (d) If we did not neglect cell death (that is, \(m \neq 0\) ), would our estimate for the time taken by the population to reach 6000 cells increase or decrease from the value given in (b)?

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ e^{x}=4 x $$ in the interval \((2,3)\) correct to six decimal places.

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

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. \(y=\sqrt{(} 2 x-1), 1 / 2 \leq x \leq 2\)

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. $$ y=x^{2}-x-4, x \in \mathbf{R} $$

\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.)$$ f(x)=\sin (x-2), 0 \leq x \leq \pi $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow-2} \frac{3 x^{2}+5 x-2}{x+2} $$

Follow Steps 1-10 to find the roots, increasing and decreasing intervals and concave up and concave down intervals of each function along with its behavior at any endpoints (or as \(x \pm \infty)\). Sketch the function, noting on your graph where any local extrema or inflection points are. Determine whether the functions have global maxima and minima, and, if so, note their location on the graph. $$ y=x^{3}-27,-5 \leq x \leq 5 $$

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ e^{x}+x=2 $$ that is correct to six decimal places.

Find the equilibria of $$x_{t+1}=\frac{2}{3}-\frac{2}{3} x_{t}^{2}, \quad t=0,1,2, \ldots$$ 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. $$ f(x)=3\left(x^{2}-x^{4}\right) $$

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. \(y=\ln \left(\frac{x}{x+1}\right), x>0\)

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. $$ y=2 x^{2}-x+3, x \in \mathbf{R} $$

\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.) $$ f(x)=\sin \frac{x}{2}, 0 \leq x \leq 2 \pi $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow-1} \frac{x+1}{x^{2}-2 x-3} $$

A rectangular field is bounded on one side by a river and on the other three sides by a fence. Find the dimensions of the field that will maximize the enclosed area if the fence has a total length of \(320 \mathrm{ft}\).

A population of cells is grown in the presence of an antibiotic; the antibiotic stresses the cells and alters their division time and their life time. For unstressed cells the division time is \(t_{b}=\) 1 hour. Initially you assume that the division time remains the same in the presence of the antibiotic, and that stressing the cells only decreases their lifetime. (a) You start a population with 2000 cells. Two hours later there are still 2000 cells present. Calculate the lifetime of the cells. (b) In fact you find in an independent measurement that the real lifetime of cells in the presence of antibiotic is \(t_{m}=4\) hours. How would you explain the data from part (a)?

Use the Newton-Raphson method to find a numerical approximation to the solution of $$ x^{2}+\ln x=0, x>0 $$ that is correct to six decimal places.

Find the equilibria of $$x_{t+1}=\frac{3}{5} x_{t}^{2}-\frac{2}{5}, \quad t=0,1,2, \ldots$$ 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. $$ f(x)=(x-1)(x+1) $$

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. \(y=x e^{-x}, 0 \leq x \leq 1\)

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. $$ y=\frac{1}{3} x^{3}-2 x^{2}+3 x+4, x \in \mathbf{R} $$

\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.) $$ f(x)=|x|,-1 \leq x \leq 1 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{3-\sqrt{2 x+9}}{2 x} $$

Find the largest possible area of a right triangle whose hypotenuse is \(4 \mathrm{~cm}\) long.

Initially you measure that a colony of bacterial cells contains 1000 cells. 2 hours later you measure the colony again, and count 2000 cells. (a) How many cells would you expect the colony to contain 4 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 4 hours after the start of the experiment? And what is the smallest possible number of cells at 4 hours?

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

Find the equilibria of $$x_{t+1}=\frac{1}{4} x_{t}^{2}+x_{t}-\frac{1}{4}, \quad t=0,1,2, \ldots$$ 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. $$ f(x)=x^{3}+x^{2}-5 x $$

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. \(y=\left|x^{2}-25\right|,-5 \leq x \leq 8\)

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. $$ y=x^{3}-5 x^{2}+8 x+2, x \in \mathbf{R} $$

\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.) $$ f(x)=(x-1)^{2}(x+1),-2 \leq x \leq 2 $$

Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2 x+4}-2}{x} $$

Suppose that \(a\) and \(b\) are the side lengths in a right triangle whose hypotenuse is \(5 \mathrm{~cm}\) long. What is the largest perimeter possible?