Chapter 4

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x^{2}-8 x+15 ;[3,5] $$

Approximate \(\sqrt{2}\) by applying Newton's Method to the equation \(x^{2}-2=0\)

Find a number in the closed interval \(\left[\frac{1}{2}, \frac{3}{2}\right]\) such that the sum of the number and its reciprocal is (a) as small as possible (b) as large as possible.

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. $$ \frac{2 x-6}{4-x} $$

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has exactly one relative extremum. (b) \(f\) is concave up on the interval \((-\infty,+\infty)\) and has no relative extrema. (c) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow+\infty\) as \(x \rightarrow+\infty\) (d) The function \(f\) has exactly two relative extrema on the interval \((-\infty,+\infty),\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow+\infty\)

In each part, sketch the graph of a function \(f\) with the stated properties, and discuss the signs of \(f^{\prime}\) and \(f^{\prime \prime} .\) (a) The function \(f\) is concave up and increasing on the interval \((-\infty,+\infty)\) (b) The function \(f\) is concave down and increasing on the interval \((-\infty,+\infty) .\) (c) The function \(f\) is concave up and decreasing on the interval \((-\infty,+\infty) .\) (d) The function \(f\) is concave down and decreasing on the interval \((-\infty,+\infty) .\)

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x^{3}-3 x^{2}+2 x ;[0,2] $$

How should two nonnegative numbers be chosen so that their sum is 1 and the sum of their squares is (a) as large as possible (b) as small as possible?

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. $$ \frac{8}{x^{2}-4} $$

In each part, sketch the graph of a continuous function \(f\) with the stated properties. (a) \(f\) has exactly one relative extremum on \((-\infty,+\infty)\) and \(f(x) \rightarrow 0\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\) (b) \(f\) has exactly two relative extrema on \((-\infty,+\infty),\) and \(f(x) \rightarrow 0 \text { as } x \rightarrow+\infty \text { and as } x \rightarrow-\infty\)

In each part, sketch the graph of a function \(f\) with the stated properties. (a) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty)\) (b) \(f\) is increasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty) .\) (c) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave up on \((0,+\infty) .\) (d) \(f\) is decreasing on \((-\infty,+\infty),\) has an inflection point at the origin, and is concave down on \((0,+\infty) .\)

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=\cos x ;[\pi / 2,3 \pi / 2] $$

Approximate \(\sqrt[3]{6}\) by applying Newton's Method to the equation \(x^{3}-6=0\).

In each part, sketch the graph of a continuous function \(f\) with the stated properties on the interval \([0,10]\) (a) \(f\) has an absolute minimum at \(x=0\) and an absolute maximum at \(x=10\). (b) \(f\) has an absolute minimum at \(x=2\) and an absolute maximum at \(x=7\). (c) \(f\) has relative minima at \(x=1\) and \(x=8,\) has relative maxima at \(x=3\) and \(x=7,\) has an absolute minimum at \(x=5,\) and has an absolute maximum at \(x=10\).

A rectangular field is to be bounded by a fence on three sides and by a straight stream on the fourth side. Find the dimensions of the field with maximum area that can be enclosed using 1000 ft of fence.

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. $$ \frac{x}{x^{2}-4} $$

(a) Use both the first and second derivative tests to show that \(f(x)=3 x^{2}-6 x+1\) has a relative minimum at \(x=1 .\) (b) Use both the first and second derivative tests to show that \(f(x)=x^{3}-3 x+3\) has a relative minimum at \(x=1\) and a relative maximum at \(x=-1\)

Verify that the hypotheses of Rolle's Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=\ln \left(4+2 x-x^{2}\right) ;[-1,3] $$

In each part, sketch the graph of a continuous function \(f\) with the stated properties on the interval \((-\infty,+\infty)\) (a) \(f\) has no relative extrema or absolute extrema. (b) \(f\) has an absolute minimum at \(x=0\) but no absolute maximum. (c) \(f\) has an absolute maximum at \(x=-5\) and an absolute minimum at \(x=5\).

The boundary of a field is a right triangle with a straight stream along its hypotenuse and with fences along its other two sides. Find the dimensions of the ficld with maximum area that can be enclosed using 1000 ft of fence.

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. $$ \frac{x^{2}}{x^{2}-4} $$

(a) Use both the first and second derivative tests to show that \(f(x)=\sin ^{2} x\) has a relative minimum at \(x=0\). (b) Use both the first and second derivative tests to show that \(g(x)=\tan ^{2} x\) has a relative minimum at \(x=0\). (c) Give an informal verbal argument to explain without calculus why the functions in parts (a) and (b) have relative minima at \(x=0 .\)

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x^{2}-x ;[-3,5] $$

Sketch a reasonable graph of \(s\) versus \(t\) for a mouse that is trapped in a narrow corridor (an \(s\) -axis with the positive direction to the right) and scurries back and forth as follows. It runs right with a constant speek and forth as for a while, then gradually slows down \(t 0.6 \mathrm{m} / \mathrm{s}\), then quickly speeds up to \(2.0 \mathrm{m} / \mathrm{s}\), then gradually slows to a stop but immediately reverses direction and quickly speeds up to \(1.2 \mathrm{m} / \mathrm{s}\).

Let $$f(x)=\left\\{\begin{array}{ll}{\frac{1}{1-x},} & {0 \leq x<1} \\ {0,} & {x=1}\end{array}\right.$$ Explain why \(f\) has a minimum value but no maximum value on the closed interval \([0,1] .\)

A rectangular plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for \(\$ 3\) a foot, while the remaining two sides will use standard fencing selling for \(\$ 2\) a foot. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a cost of \(\$ 6000 ?\)

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. $$ \frac{x^{2}}{x^{2}+4} $$

(a) Show that both of the functions \(f(x)=(x-1)^{4}\) and \(g(x)=x^{3}-3 x^{2}+3 x-2\) have stationary points at \(x=1 .\) (b) What does the second derivative test tell you about the nature of these stationary points? (c) What does the first derivative test tell you about the nature of these stationary points?

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x^{3}+x-4 ;[-1,2] $$

Let $$f(x)=\left\\{\begin{array}{ll}{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. $$ \frac{\left(x^{2}-1\right)^{2}}{x^{4}+1} $$

(a) Show that \(f(x)=1-x^{5}\) and \(g(x)=3 x^{4}-8 x^{3}\) both have stationary points at \(x=0\) (b) What does the second derivative test tell you about the nature of these stationary points? (c) What does the first derivative test tell you about the nature of these stationary points?

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=\sqrt{x+1} ;[0,3] $$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. \(f(x)=4 x^{2}-12 x+10 ;[1,2]\)

Locate the critical points and identify which critical points are stationary points. $$ f(x)=4 x^{4}-16 x^{2}+17 $$

Verify that the hypotheses of the Mean-Value Theorem are satisfied on the given interval, and find all values of \(c\) in that interval that satisfy the conclusion of the theorem. $$ f(x)=x-\frac{1}{x} ;[3,4] $$

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. \(f(x)=8 x-x^{2} ;[0,6]\)

A rectangle has its two lower corners on the \(x\) -axis and its two upper corners on the curve \(y=16-x^{2} .\) For all such rectangles, what are the dimensions of the one with largest area?

Locate the critical points and identify which critical points are stationary points. $$ f(x)=3 x^{4}+12 x $$

(a) Find an interval \([a, b]\) on which $$ f(x)=x^{4}+x^{3}-x^{2}+x-2 $$ satisfies the hypotheses of Rolle's Theorem. (b) Generate the graph of \(f^{\prime}(x),\) and use it to make rough estimates of all values of \(c\) in the interval obtained in part (a) that satisfy the conclusion of Rolle's Theorem. (c) Use Newton's Method to improve on the rough estimates obtained in part (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. \(x^{4}+x^{2}-4=0 ; x<0\)

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

Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius \(10 .\)

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. $$ \frac{4}{x^{2}}-\frac{2}{x}+3 $$

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

Let \(f(x)=x^{3}-4 x\) (a) Find the equation of the secant line through the points \((-2, f(-2))\) and \((1, f(1))\). (b) Show that there is only one point \(c\) in the interval \((-2,1)\) that satisfies the conclusion of the Mean-Value Theorem for the secant line in part (a). (c) Find the equation of the tangent line to the graph of \(f\) at the point \((c, f(c)) .\) (d) Use a graphing utility to generate the secant line in part (a) and the tangent line in part (c) in the same coordinate system, and confirm visually that the two lines seem parallel.

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. \(x^{5}-5 x^{3}-2=0 ; x >0\)

Determine whether the statement is true or false. Explain your answer. Velocity is the derivative of position with respect to time.

Find the absolute maximum and minimum values of \(f\) on the given closed interval, and state where those values occur. \(f(x)=2 x^{3}+3 x^{2}-12 x ;[-3,2]\)

Find the point \(P\) in the first quadrant on the curve \(y=x^{-2}\) such that a rectangle with sides on the coordinate axes and a vertex at \(P\) has the smallest possible perimeter.