Chapter 4

Suppose that a runner has velocity 15 mph for 20 minutes, velocity 18 mph for 30 minutes, velocity 16 mph for 10 minutes and velocity 12 mph for 40 minutes. Find the distance run.

Evaluate the indicated integral. $$\int \frac{1}{x \sqrt{x^{4}-1}} d x$$

Evaluate the integral. $$\int_{e}^{e^{2}} \frac{1}{x \ln x} d x$$

Use (a) Trapezoidal Rule and (b) Simpson's Rule to estimate \(\int_{0}^{2} f(x) d x\) from the given data. $$\begin{array}{|l|l|l|l|l|r|} \hline x & 0.0 & 0.25 & 0.5 & 0.75 & 1.0 \\ \hline f(x) & 1.0 & 0.6 & 0.2 & -0.2 & -0.4 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|} \hline x & 1.25 & 1.5 & 1.75 & 2.0 \\ \hline f(x) & 0.4 & 0.8 & 1.2 & 2.0 \\ \hline \end{array}$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{2}^{x^{2}+1} \sin t d t\)

Show that for left-endpoint evaluation on the interval \([a, b]\) with each sub interval of length \(\Delta x=(b-a) / n,\) the evaluation points are \(c_{i}=a+(i-1) \Delta x,\) for \(i=1,2, \ldots, n\)

Use the Integral Mean Value Theorem to estimate the value of the integral. $$\int_{0}^{1 / 2} e^{-x^{2}} d x$$

Find the general antiderivative. $$\int x^{2 / 3}\left(x^{-4 / 3}-3\right) d x$$

Suppose that a runner has velocity 12 mph for 20 minutes, velocity 14 mph for 30 minutes, velocity 18 mph for 10 minutes and velocity 15 mph for 40 minutes. Find the distance run.

Evaluate the definite integral. $$\int_{0}^{2} x \sqrt{x^{2}+1} d x$$

Graph the function. $$y=\ln (x-2)$$

The table gives the measurements (in feet) of the width of a plot of land at 10 -foot intervals. Estimate the area of the plot. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline x & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline f(x) & 56 & 54 & 58 & 62 & 58 & 58 & 62 \\ \hline \end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 70 & 80 & 90 & 100 & 110 & 120 \\ \hline f(x) & 56 & 52 & 48 & 40 & 32 & 22 \\ \hline \end{array}$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{x}^{-1} \ln \left(t^{2}+1\right) d t\)

Use the Integral Mean Value Theorem to estimate the value of the integral. $$\int_{0}^{2} \sqrt{2 x^{2}+1} d x$$

One of the two antiderivatives can be determined using basic algebra and the derivative formulas we have presented. Find the antiderivative of this one and label the other \(^{\text {*. }} \mathbf{N} / \mathbf{A}^{\mathbf{\prime}}\) $$\text { (a) } \int \sqrt{x^{3}+4} d x$$ $$\text { (b) } \int(\sqrt{x^{3}}+4) d x$$

The table shows the velocity of a projectile at various times. Estimate the distance traveled. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { time (s) } & 0 & 0.25 & 0.5 & 0.75 & 1.0 & 1.25 & 1.5 & 1.75 & 2.0 \\ \hline \text { velocity (ft/s) } & 120 & 116 & 113 & 110 & 108 & 106 & 104 & 103 & 102 \\ \hline \end{array}$$

Evaluate the definite integral. $$\int_{1}^{3} x \sin \left(\pi x^{2}\right) d x$$

Graph the function. $$y=\ln (3 x+5)$$

The table gives the measurements (in feet) of the width of a plot of land at 10 -foot intervals. Estimate the area of the plot. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline x & 0 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline f(x) & 26 & 30 & 28 & 22 & 28 & 32 & 30 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 70 & 80 & 90 & 100 & 110 & 120 \\ \hline f(x) & 33 & 31 & 28 & 30 & 32 & 22 \\ \hline \end{array}$$

find the derivative \(f^{\prime}(x)\) \(f(x)=\int_{x}^{2} \sec t d t\)

Use the Integral Mean Value Theorem to estimate the value of the integral. $$\int_{-1}^{1} \frac{3}{x^{3}+2} d x$$

Evaluate the definite integral. $$\int_{-1}^{1} \frac{x}{\left(x^{2}+1\right)^{2}} d x$$

Graph the function. $$y=\ln \left(x^{2}+1\right)$$

The velocity of an object at various times is given. Use the data to estimate the distance traveled. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 40 & 42 & 40 & 44 & 48 & 50 & 46 \\ \hline \end{array}$$ $$\begin{array}{|l|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 46 & 42 & 44 & 40 & 42 & 42 \\ \hline \end{array}$$

Find an equation of the tangent line at the given value of \(x\) $$y=\int_{0}^{x} \sin \sqrt{t^{2}+\pi^{2}} d t, x=0$$

Economists use a graph called the Lorentz curve to describe how equally a given quantity is distributed in a given population. For example, the gross domestic product (GDP) varies considerably from country to country. The accompanying data from the Energy Information Administration show percentages for the 100 top-GDP countries in the world in 2001 , arranged in order of increasing GDP. The data indicate that the first 10 (lowest \(10 \%\) ) countries account for only \(0.2 \%\) of the world's total GDP; the first 20 countries account for \(0.4 \%\) and so on. The first 99 countries account for \(73.6 \%\) of the total GDP. What percentage does country \(\%\) 100 (the United States) produce? The Lorentz curve is a plot of \(y\) versus \(x\). Graph the Lorentz curve for these data. Estimate the area between the curve and the \(x\) -axis. (Hint: Notice that the \(x\) -values are not equally spaced. You will need to decide how to handle this. Depending on your choice, your answer may not exactly match the back of the book; this is OK!) $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 \\ \hline y & 0.002 & 0.004 & 0.008 & 0.014 & 0.026 & 0.048 & 0.085 \\ \hline \end{array}$$ $$\begin{array}{|c|c|c|c|c|c|c|} \hline x & 0.8 & 0.9 & 0.95 & 0.98 & 0.99 & 1.0 \\ \hline y & 0.144 & 0.265 & 0.398 & 0.568 & 0.736 & 1.0 \\ \hline \end{array}$$

Find a value of \(c\) that satisfies the conclusion of the Integral Mean Value Theorem. $$\int_{0}^{2} 3 x^{2} d x(=8)$$

One of the two antiderivatives can be determined using basic algebra and the derivative formulas we have presented. Find the antiderivative of this one and label the other \(^{\text {*. }} \mathbf{N} / \mathbf{A}^{\mathbf{\prime}}\) $$\text { (a) } \int 2 \sec x \, d x$$ $$\text { (b) } \int \sec ^{2} x d x$$

Use mathematical induction to prove that \(\sum_{i=1}^{n} i^{3}=\frac{n^{2}(n+1)^{2}}{4}\) for all integers \(n \geq 1 .\)

Evaluate the definite integral. $$\int_{0}^{2} x^{2} e^{x^{3}} d x$$

Graph the function. $$y=\ln \left(x^{3}+1\right)$$

The velocity of an object at various times is given. Use the data to estimate the distance traveled. $$\begin{array}{|l|r|r|r|r|r|r|r|} \hline t(\mathrm{s}) & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 26 & 30 & 28 & 30 & 28 & 32 & 30 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 14 & 16 & 18 & 20 & 22 & 24 \\ \hline v(t)(\mathrm{ft} / \mathrm{s}) & 33 & 31 & 28 & 30 & 32 & 32 \\ \hline \end{array}$$

Find a value of \(c\) that satisfies the conclusion of the Integral Mean Value Theorem. $$\int_{-1}^{1}\left(x^{2}-2 x\right) d x\left(=\frac{2}{3}\right)$$

One of the two antiderivatives can be determined using basic algebra and the derivative formulas we have presented. Find the antiderivative of this one and label the other \(^{\text {*. }} \mathbf{N} / \mathbf{A}^{\mathbf{\prime}}\) $$\text { (a) } \int\left(\frac{1}{x^{2}}-1\right) d x$$ $$\text { (b) } \int \frac{1}{x^{2}-1} d x$$

Use mathematical induction to prove that \(\sum_{i=1}^{n} i^{5}=\frac{n^{2}(n+1)^{2}\left(2 n^{2}+2 n-1\right)}{12}\) for all integers \(n \geq 1.\)

Evaluate the definite integral. $$\int_{0}^{2} \frac{e^{x}}{1+e^{2 x}} d x$$

Graph the function. $$y=x \ln x$$

The data come from a pneumotachograph, which measures air flow through the throat (in liters per second). The integral of the air flow equals the volume of air exhaled. Estimate this volume. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 \\ \hline f(t)(1 / s) & 0 & 0.2 & 0.4 & 1.0 & 1.6 & 2.0 & 2.2 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 \\ \hline f(t)(1 / \mathrm{s}) & 2.0 & 1.6 & 1.2 & 0.6 & 0.2 & 0 \\ \hline \end{array}$$

Find an equation of the tangent line at the given value of \(x\) $$y=\int_{2}^{x} \cos \left(\pi t^{3}\right) d t, x=2$$

Use the following definitions. The upper sum of \(f\) on \(P\) is given by \(U(P, f)=\sum_{i=1}^{n} f\left(c_{i}\right) \Delta x,\) where \(f\left(c_{i}\right)\) is the maximum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right] .\) Similarly, the lower sum of \(f\) on \(P\) is given by \(L(P, f)=\sum_{i=1}^{m} f\left(d_{i}\right) \Delta x,\) where \(f\left(d_{i}\right)\) is the minimum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right]\) Compute the upper sum and lower sum of \(f(x)=x^{2}\) on [0,2] for the regular partition with \(n=4\)

Compute the average value of the function on the given interval. $$f(x)=2 x+1,[0,4]$$

Use the formulas in exercises 33 and 34 to compute the sums. $$\sum_{i=1}^{10}\left(i^{3}-3 i+1\right)$$

Evaluate the definite integral. $$\int_{0}^{\pi^{2}} \frac{\cos \sqrt{x}}{\sqrt{x}} d x$$

Graph the function. $$y=x^{2} \ln x$$

The data come from a pneumotachograph, which measures air flow through the throat (in liters per second). The integral of the air flow equals the volume of air exhaled. Estimate this volume. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 \\ \hline f(t)(1 / \mathrm{s}) & 0 & 0.1 & 0.4 & 0.8 & 1.4 & 1.8 & 2.0 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|l|} \hline t(\mathrm{s}) & 1.4 & 1.6 & 1.8 & 2.0 & 2.2 & 2.4 \\ \hline f(t)(1 / \mathrm{s}) & 2.0 & 1.6 & 1.0 & 0.6 & 0.2 & 0 \\ \hline \end{array}$$

Use the following definitions. The upper sum of \(f\) on \(P\) is given by \(U(P, f)=\sum_{i=1}^{n} f\left(c_{i}\right) \Delta x,\) where \(f\left(c_{i}\right)\) is the maximum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right] .\) Similarly, the lower sum of \(f\) on \(P\) is given by \(L(P, f)=\sum_{i=1}^{m} f\left(d_{i}\right) \Delta x,\) where \(f\left(d_{i}\right)\) is the minimum of \(f\) on the sub interval \(\left[x_{i-1}, x_{i}\right]\) Compute the upper sum and lower sum of \(f(x)=x^{2}\) on [-2,2] for the regular partition with \(n=8\)

Compute the average value of the function on the given interval. $$f(x)=x^{2}+2 x,[0,1]$$

Use the formulas in exercises 33 and 34 to compute the sums. $$\sum_{i=1}^{20}\left(i^{3}+2 i\right)$$

Evaluate the definite integral. $$\int_{\pi / 4}^{\pi / 2} \cot x d x$$

Identify all local extrema of \(f(x)=\int_{0}^{x}\left(t^{2}-3 t+2\right) d t\)