Chapter 4

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=4 x+9 / x $$

Solve each of the maximum-minimum problems. Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. An open-topped rectangular planter is to hold \(3 \mathrm{~m}^{3}\). Its concrete base is square. Each brick side is rectangular. If the unit cost of brick is twelve times that of concrete, what dimensions result in the cheapest material cost?

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=x /\left(x^{2}+3\right) $$

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=e^{x}-x $$

Variables \(x\) and \(y,\) which depend on \(t,\) are related by a given equation. A point \(P_{0}\) on the graph of that equation is also given, as is one of the following two values: $$v_{0}=\left.\frac{d x}{d t}\right|_{P_{0}} \quad \text { or } \quad s_{0}=\left.\frac{d y}{d t}\right|_{P_{0}}$$ Find the other. $$ 2 x^{3}-x^{2} y+3 y=13, \quad P_{0}=(2,3), \quad v_{0}=-2 $$

Calculate the indefinite integral. $$ \int(3 \sin (7 x)+7 \sin (3 x)) d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow \pi / 2} \tan (2 x) / \cot (x)\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=x-\sqrt{x} $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=x / \sqrt{x^{2}+1} $$

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=x e^{x} $$

Variables \(x\) and \(y,\) which depend on \(t,\) are related by a given equation. A point \(P_{0}\) on the graph of that equation is also given, as is one of the following two values: $$v_{0}=\left.\frac{d x}{d t}\right|_{P_{0}} \quad \text { or } \quad s_{0}=\left.\frac{d y}{d t}\right|_{P_{0}}$$ Find the other. $$ x^{2}+x \exp (y)+y=6, \quad P_{0}=(2,0), \quad s_{0}=-1 $$

Calculate the indefinite integral. $$ \int \sec ^{2}(8 x) d x $$

A Babylonian approximation to \(\sqrt{c}\) from the second millennium B.C. consists of starting with a first estimate \(x_{1}\) and then computing the subsequent approximations that are defined iteratively by $$ x_{j+1}=\frac{1}{2}\left(x_{j}+\frac{c}{x_{j}}\right) $$ Show that this ancient algorithm is the Newton-Raphson Method applied to \(f(x)=x^{2}-c\). Of course there was no notion of derivative when this algorithm was discovered. Considering that \(x_{j+1}\) is the average of \(x_{j}\) and \(c / x_{j}\), what might Babylonian mathematicians have had in mind when they devised this algorithm?

In each of Exercises \(17-24,\) apply l'Hôpital's Rule repeatedly (when needed) to evaluate the given limit, if it exists. \(\lim _{x \rightarrow \infty} x^{2} / e^{3 x}\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=3 x+2 \sin (x) $$

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=x(x+1)^{1 / 5} $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=\left(x^{2}-1\right) /\left(x^{2}+1\right) $$

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=e^{x} / x $$

Calculate the indefinite integral. $$ \int \csc (2 x) \cot (2 x) d x $$

Define $$ g(x)=\left\\{\begin{array}{cc} (x-4)^{1 / 2} & \text { if } x \geq 4 \\ -(4-x)^{1 / 2} & \text { if } x<4 \end{array}\right. $$ Prove that, for any initial guess other than \(x=4,\) the Newton-Raphson Method will not converge.

Apply l'Hôpital's Rule repeatedly (when needed) to evaluate the given limit, if it exists. \(\lim _{x \rightarrow \infty} x^{3} / e^{2 x}\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=2 \cos ^{2}(x)+3 x $$

Solve each of the maximum-minimum problems. Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. What is the rectangle (with sides parallel to the axes) of greatest area that can be inscribed in the ellipse \(x^{2}+4 y^{2}=16 ?\)

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=|x| /(x+1) $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=(x+1)^{2} / x $$

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=x \ln (x) $$

Variables \(x\) and \(y\) are functions of a parameter \(t\) and are related by a given equation. A point \(P_{0}\) on the graph of that equation is also given, as is one of the following two values: $$ v_{0}=\left.\frac{d x}{d t}\right|_{P_{0}} \quad \text { or } \quad s_{0}=\left.\frac{d y}{d t}\right|_{P_{0}} $$ Find the other value. Also, referring to the Insight that follows Example \(2,\) find the tangent line to the graph of the given equation at \(P_{0}\). $$ 2 x^{3}+y^{3}=8, \quad P_{0}=(2,-2), \quad v_{0}=-1 $$

Calculate the indefinite integral. $$ \int(3 x-2)^{3} d x $$

Apply l'Hôpital's Rule repeatedly (when needed) to evaluate the given limit, if it exists. \(\lim _{x \rightarrow 0} \frac{\sin ^{2}(x)}{x^{2}}\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=e^{x}-x $$

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=x^{1 / 3}(x-2) $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=\cos (x)+x, 0

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=\ln (x) / x $$

Calculate the indefinite integral. $$ \int x^{2}(\sqrt{x}+1)^{2} d x $$

Apply l'Hôpital's Rule repeatedly (when needed) to evaluate the given limit, if it exists. \(\lim _{x \rightarrow 1} \frac{(x-1)^{3}}{\ln ^{2}(x)}\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=x e^{-x} $$

Solve each of the maximum-minimum problems. Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. Assume that if the price of a certain book is \(p\) dollars, then it will sell \(x\) copies where \(x=7000 \cdot(1-p / 35)\). Suppose that the dollar cost of producing those \(x\) copies is \(15000+2.5 x .\) Finally, assume that the company will not sell this book for more than \(\$ 35 .\) Determine the price for the book that will maximize profit.

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=\sqrt{x} /(\sqrt{x}-1) $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=x / 2-\sin (x), 0

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=\sqrt{x}+2 / \sqrt{x} $$

Variables \(x\) and \(y\) are functions of a parameter \(t\) and are related by a given equation. A point \(P_{0}\) on the graph of that equation is also given, as is one of the following two values: $$ v_{0}=\left.\frac{d x}{d t}\right|_{P_{0}} \quad \text { or } \quad s_{0}=\left.\frac{d y}{d t}\right|_{P_{0}} $$ Find the other value. Also, referring to the Insight that follows Example \(2,\) find the tangent line to the graph of the given equation at \(P_{0}\). $$ 2 x-\sqrt{x}-\sqrt{y}=3, \quad P_{0}=(4,9), \quad s_{0}=2 $$

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=6 x^{2}, F(1)=3, c=0 $$

Apply l'Hôpital's Rule repeatedly (when needed) to evaluate the given limit, if it exists. \(\lim _{x \rightarrow 1} \frac{(x-1)^{2}}{\cos ^{2}(\pi x / 2)}\)

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=x^{2} e^{-x} $$

In each of Exercises \(21-24,\) a cost function \(c\) and \(a\) form of the demand equation are given. Calculate the sales level \(x\) that maximizes profits. \(C(x)=1000+2 x, D^{-1}(x)=1602-8 x\)

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=\sqrt{x} /(x+1) $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=\tan (x / 2),-\pi

In each of Exercises \(7-22,\) use Fermat's Theorem to locate each \(c\) for which \(f(c)\) is a candidate extreme value of the given function \(f\) $$ f(x)=x^{3 / 2}-9 x^{1 / 2} $$

Variables \(x\) and \(y\) are functions of a parameter \(t\) and are related by a given equation. A point \(P_{0}\) on the graph of that equation is also given, as is one of the following two values: $$ v_{0}=\left.\frac{d x}{d t}\right|_{P_{0}} \quad \text { or } \quad s_{0}=\left.\frac{d y}{d t}\right|_{P_{0}} $$ Find the other value. Also, referring to the Insight that follows Example \(2,\) find the tangent line to the graph of the given equation at \(P_{0}\). $$ 2 x^{2}-x y^{2}+\ln (y / 2)=6, \quad P_{0}=(3,2), \quad v_{0}=-2 $$

Compute \(F(c)\) from the given information. $$ F^{\prime}(x)=2 x+3, F(1)=2, c=-1 $$