Chapter 4

Use a graphing utility to determine how many solutions the equation has, and then use Newton's Method to approximate the solution that satisfies the stated condition. $$2 \cos x=x ; x>0$$

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$\frac{(3 x+1)^{2}}{(x-1)^{2}}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=\frac{3 x}{\sqrt{4 x^{2}+1}} ;[-1,1]$$

A rectangular area of \(3200 \mathrm{ft}^{2}\) is to be fenced off. Two opposite sides will use fencing costing 1 dollars per foot and the remaining sides will use fencing costing 2 dollars per foot. Find the dimensions of the rectangle of least cost.

Locate the critical points and identify which critical points are stationary points. $$f(x)=\sqrt[3]{x^{2}-25}$$

Determine whether the statement is true or false. Explain your answer. If \(f\) is continuous on a closed interval \([a, b]\) and differentiable on \((a, b),\) then there is a point between \(a\) and \(b\) at which the instantaneous rate of change of \(f\) matches the average rate of change of \(f\) over \([a, b]\)

Use a graphing utility to determine how many solutions the equation has, and then use Newton's Method to approximate the solution that satisfies the stated condition. $$\sin x=x^{2} ; x>0$$

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$3+\frac{x+1}{(x-1)^{4}}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=\left(x^{2}+x\right)^{2 / 3} ;[-2,3]$$

Show that among all rectangles with perimeter \(p\), the square has the maximum area.

Locate the critical points and identify which critical points are stationary points. $$f(x)=x^{2}(x-1)^{2 / 3}$$

Use a graphing utility to determine how many solutions the equation has, and then use Newton's Method to approximate the solution that satisfies the stated condition. $$x-\tan x=0 ; \pi / 2

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$\frac{x^{2}+x}{1-x^{2}}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=x-2 \sin x ;[-\pi / 4, \pi / 2]$$

Locate the critical points and identify which critical points are stationary points. $$f(x)=|\sin x|$$

Show that among all rectangles with area \(A\), the square has the minimum perimeter.

Determine whether the statement is true or false. Explain your answer. One application of the Mean-Value Theorem is to prove that a function with positive derivative on an interval must be increasing on that interval.

True-False Assume that \(f\) is differentiable everywhere. Determine whether the statement is true or false. Explain your answer. If \(f^{\prime}\) is increasing on [0,1] and \(f^{\prime}\) is decreasing on [1,2] then \(f\) has an inflection point at \(x=1\)

Use a graphing utility to determine how many solutions the equation has, and then use Newton's Method to approximate the solution that satisfies the stated condition. $$1+e^{x} \sin x=0 ; \pi / 2

Give a graph of the rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label point(s), if any, where the graph crosses a horizontal asymptote. Check your work with a graphing utility. $$\frac{x^{2}}{1-x^{3}}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=\sin x-\cos x ;[0, \pi]$$

Locate the critical points and identify which critical points are stationary points. $$f(x)=\sin |x|$$

A wire of length 12 in can be bent into a circle, bent into a square, or cut into two pieces to make both a circle and a square. How much wire should be used for the circle if the total area enclosed by the figure(s) is to be (a) a maximum (b) a minimum?

Let \(f(x)=\tan x\) (a) Show that there is no point \(c\) in the interval \((0, \pi)\) such that \(f^{\prime}(c)=0,\) even though \(f(0)=f(\pi)=0\) (b) Explain why the result in part (a) does not contradict Rolle's Theorem.

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up, (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=x^{2}-3 x+8$$

Use a graphing utility to determine the number of times the curves intersect; and then apply Newton's Method, where needed, to approximate the \(x\) -coordinates of all intersections. $$y=x^{3} \text { and } y=1-x$$

The function \(s(t)\) describes the position of a particle moving along a coordinate line, where \(s\) is in meters and \(t\) is in seconds. (a) Make a table showing the position, velocity, and acceleration to two decimal places at times \(t=1,2,3,4,5\) (b) At each of the times in part (a), determine whether the particle is stopped; if it is not, state its direction of motion. (c) At each of the times in part (a), determine whether the particle is speeding up, slowing down, or neither. $$s(t)=\sin \frac{\pi t}{4}$$

In each part, make a rough sketch of the graph using asymptotes and appropriate limits but no derivatives. Compare your graph to that generated with a graphing utility. $$\text { (a) } y=\frac{3 x^{2}-8}{x^{2}-4}$$ $$\text { (b) } y=\frac{x^{2}+2 x}{x^{2}-1}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=1+\left|9-x^{2}\right| ;[-5,1]$$

Assume that \(f\) is continuous everywhere. Determine whether the statement is true or false. Explain your answer. If \(f\) has a relative maximum at \(x=1,\) then \(f(1) \geq f(2)\)

A rectangle \(R\) in the plane has corners at \((\pm 8,\pm 12),\) and a 100 by 100 square \(S\) is positioned in the plane so that its sides are parallel to the coordinate axes and the lower left corner of \(S\) is on the line \(y=-3 x .\) What is the largest possible area of a region in the plane that is contained in both \(R\) and \(S ?\)

Let \(f(x)=x^{2 / 3}, a=-1,\) and \(b=8\) (a) Show that there is no point \(c\) in \((a, b)\) such that $$ f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} $$ (b) Explain why the result in part (a) does not contradict the Mean-Value Theorem.

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up, (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=5-4 x-x^{2}$$

Use a graphing utility to determine the number of times the curves intersect; and then apply Newton's Method, where needed, to approximate the \(x\) -coordinates of all intersections. $$y=\sin x \text { and } y=x^{3}-2 x^{2}+1$$

The function \(s(t)\) describes the position of a particle moving along a coordinate line, where \(s\) is in meters and \(t\) is in seconds. (a) Make a table showing the position, velocity, and acceleration to two decimal places at times \(t=1,2,3,4,5\) (b) At each of the times in part (a), determine whether the particle is stopped; if it is not, state its direction of motion. (c) At each of the times in part (a), determine whether the particle is speeding up, slowing down, or neither. $$s(t)=t^{4} e^{-t}, \quad t \geq 0$$

In each part, make a rough sketch of the graph using asymptotes and appropriate limits but no derivatives. Compare your graph to that generated with a graphing utility. $$\text { (a) } y=\frac{2 x-x^{2}}{x^{2}+x-2}$$ $$\text { (b) } y=\frac{x^{2}}{x^{2}-x-2}$$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. $$f(x)=|6-4 x| ;[-3,3]$$

Assume that \(f\) is continuous everywhere. Determine whether the statement is true or false. Explain your answer. If \(f\) has a relative maximum at \(x=1,\) then \(x=1\) is a critical point for \(f\).

(a) Show that if \(f\) is differentiable on \((-\infty,+\infty),\) and if \(y=f(x)\) and \(y=f^{\prime}(x)\) are graphed in the same coordinate system, then between any two \(x\) -intercepts of \(f\) there is at least one \(x\) -intercept of \(f^{\prime}\) (b) Give some examples that illustrate this.

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up, (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=(2 x+1)^{3}$$

Use a graphing utility to determine the number of times the curves intersect; and then apply Newton's Method, where needed, to approximate the \(x\) -coordinates of all intersections. $$y=x^{2} \text { and } y=\sqrt{2 x+1}$$

The function \(s(t)\) describes the position of a particle moving along a coordinate line, where \(s\) is in feet and \(t\) is in seconds. (a) Find the velocity and acceleration functions. (b) Find the position, velocity, speed, and acceleration at time \(t=1\) (c) At what times is the particle stopped? (d) When is the particle speeding up? Slowing down? (e) Find the total distance traveled by the particle from time \(t=0\) to time \(t=5\) $$s(t)=t^{3}-3 t^{2}, \quad t \geq 0$$

Show that \(y=x+3\) is an oblique asymptote of the graph of \(f(x)=x^{2} /(x-3) .\) Sketch the graph of \(y=f(x)\) showing this asymptotic behavior.

Determine whether the statement is true or false. Explain your answer. If a function \(f\) is continuous on \([a, b],\) then \(f\) has an absolute maximum on \([a, b]\)

Find: (a) the intervals on which \(f\) is increasing, (b) the intervals on which \(f\) is decreasing, (c) the open intervals on which \(f\) is concave up, (d) the open intervals on which \(f\) is concave down, and (e) the \(x\) -coordinates of all inflection points. $$f(x)=5+12 x-x^{3}$$

Use a graphing utility to determine the number of times the curves intersect; and then apply Newton's Method, where needed, to approximate the \(x\) -coordinates of all intersections. $$y=\frac{1}{8} x^{3}-1 \text { and } y=\cos x-2$$

The function \(s(t)\) describes the position of a particle moving along a coordinate line, where \(s\) is in feet and \(t\) is in seconds. (a) Find the velocity and acceleration functions. (b) Find the position, velocity, speed, and acceleration at time \(t=1\) (c) At what times is the particle stopped? (d) When is the particle speeding up? Slowing down? (e) Find the total distance traveled by the particle from time \(t=0\) to time \(t=5\) $$s(t)=t^{4}-4 t^{2}+4, \quad t \geq 0$$

Show that \(y=3-x^{2}\) is a curvilinear asymptote of the graph of \(f(x)=\left(2+3 x-x^{3}\right) / x .\) Sketch the graph of \(y=\) \(f(x)\) showing this asymptotic behavior.

Assume that \(f\) is continuous everywhere. Determine whether the statement is true or false. Explain your answer. If \(p(x)\) is a polynomial such that \(p^{\prime}(x)\) has a simple root at \(x=1,\) then \(p\) has a relative extremum at \(x=1\)

A rectangular page is to contain 42 square inches of printable area. The margins at the top and bottom of the page are each 1 inch, one side margin is 1 inch, and the other side margin is 2 inches. What should the dimensions of the page be so that the least amount of paper is used?