Chapter 4

Calculate the indefinite integral. $$ \int\left(x^{2}-5 x\right) d x $$

In each of Exercises \(1-16,\) use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow 0}\left(1-e^{x}\right) / x\)

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. $$ f(x)=x^{3}-3 x^{2}-9 x+7 $$

Solve each of the maximum-minimum problems in Exercises \(1-20 .\) Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. A rectangle is to have perimeter \(100 \mathrm{~m}\). What dimensions will give it the greatest area?

In each of Exercises \(1-36,\) 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. $$ f(x)=x-4 \sqrt{x} $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=(x-2)^{2}+3 $$

The variable \(y\) is given as a function of \(x\), which depends on \(t\). The values \(x_{0}\) and \(v_{0}\) of, respectively, \(x\) and \(d x / d t\) are given at a value \(t_{0}\) of \(t\). Use this data to find \(d y / d t\) at \(t_{0}\). $$ y=x^{3}, \quad x_{0}=2, \quad \quad v_{0}=5 $$

Calculate the indefinite integral. $$ \int(3 \sin (x)-5 \cos (x)+1) d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow+\infty} \ln (x) / \sqrt{x}\)

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. $$ f(x)=x^{3}(x-8) $$

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 positive number plus its reciprocal gives the least sum?

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. $$ f(x)=1 / x-1 / x^{2} $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=2-4(x-6)^{2} $$

Calculate the indefinite integral. $$ \int e^{x} d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow 5} \frac{\ln (x / 5)}{x-5}\)

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

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 first quadrant point on the curve \(x \mathrm{y}^{2}=1\) is closest to the origin?

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. $$ f(x)=\sqrt{x}+3 / \sqrt{x} $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=1 /\left(x^{2}+1\right) $$

The variable \(y\) is given as a function of \(x\), which depends on \(t\). The values \(x_{0}\) and \(v_{0}\) of, respectively, \(x\) and \(d x / d t\) are given at a value \(t_{0}\) of \(t\). Use this data to find \(d y / d t\) at \(t_{0}\). $$ y=\cos (x), \quad x_{0}=\pi / 6, \quad v_{0}=-2 $$

Calculate the indefinite integral. $$ \int \sec (x) \tan (x) d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow 0} \frac{x+\sin (5 x)}{x-3 \sin (4 x)}\)

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. $$ f(x)=4 x^{3}-6 x^{2}+8 $$

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. $$ f(x)=x^{5}-6 x^{4}+8 x^{3} $$

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. Find the longest vertical chord between the curves \(y=\) \(x^{3} / 3-3 x^{2} / 2+2 x-3 / 4\) and \(y=3 x^{3} / 4-15 x^{2}+5 x, 0 \leq x \leq 3\).

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. $$ f(x)=x \sqrt{x-2} $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=1 /\left((x-2)^{2}+4\right) $$

Calculate the indefinite integral. $$ \int \sqrt{x+2} d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow \pi / 2} \frac{\ln (\sin (x))}{(\pi-2 x)^{2}}\)

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. $$ f(x)=3 x^{4}+20 x^{3}+36 x^{2} $$

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. $$ f(x)=x /\left(x^{2}+4\right) $$

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. A box (with no lid) is to be constructed from a sheet of cardboard by cutting squares from the corners and folding up the sides (Figure 11 ). If the original sheet of cardboard measures 20 inches by 20 inches, then what should be the size of the squares removed to maximize the volume of the resulting box?

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. $$ f(x)=x^{2}+54 / x $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=3 \sin (4 x) $$

The variable \(y\) is given as a function of \(x\), which depends on \(t\). The values \(x_{0}\) and \(v_{0}\) of, respectively, \(x\) and \(d x / d t\) are given at a value \(t_{0}\) of \(t\). Use this data to find \(d y / d t\) at \(t_{0}\). $$ y=x \ln (2 x), \quad x_{0}=1 / 2, \quad v_{0}=3 $$

Calculate the indefinite integral. $$ \int\left(x^{2}-x^{-2}+x^{1 / 2}-x^{-1 / 2}\right) d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow 0} \frac{\cos (x)-1}{e^{x}-1}\)

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. $$ f(x)=(x+1) /(x-1) $$

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. Of all right triangles with hypotenuse \(100 \mathrm{~cm}\), which has greatest area?

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. $$ f(x)=1 /\left(x^{2}+1\right) $$

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=\exp (-|x|) $$

The variable \(y\) is given as a function of \(x\), which depends on \(t\). The values \(x_{0}\) and \(v_{0}\) of, respectively, \(x\) and \(d x / d t\) are given at a value \(t_{0}\) of \(t\). Use this data to find \(d y / d t\) at \(t_{0}\). $$ y=x^{3 / 2}-\exp (-2 x), \quad x_{0}=1, \quad \quad v_{0}=4 $$

Calculate the indefinite integral. $$ \int\left(\frac{x^{2}+x^{-3}}{x^{4}}\right) d x $$

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. $$ f(x)=-x /\left(x^{2}+25\right) $$

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. A line with negative slope passes through the point (1,2) and has \(x\) -intercept \(a\) and \(y\) -intercept \(b\). For what slope is the product \(a b\) minimized?

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. $$ f(x)=x^{3}+9 x^{2}-21 x+15 $$

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 \(f\) $$ f(x)=2 x^{2}-24 x+36 $$

Calculate the indefinite integral. $$ \int x^{3 / 2}\left(x^{-3}-4 x^{-2}+2 x^{-1}\right) d x $$

Use l'Hôpital's Rule to find the limit, if it exists. \(\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x}\)

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. $$ f(x)=x /(x+1) $$