Chapter 7.2, Problem 78E

a.

To determine

To explain: how can use the derivative of F(x) + C to confirm the integration is correct.

Answer to Problem 78E

If the derivative of F(x) + C is equal to f(x) then the integration was performed correctly.

Explanation of Solution

Given information: Suppose that ${\displaystyle \int \mathrm{f}(\mathrm{x})\mathrm{d}\mathrm{x}=\mathrm{F}(\mathrm{x})+\mathrm{C}}$ .

Calculation:

If the derivative of F(x) + C is equal to f(x) then the integration was performed correctly.

b.

To determine

To explain: how can use a slope field of f and the graph of y = F(x) to support evaluation of the integral.

Answer to Problem 78E

The slope field generated by f ( x ) should generate all curves in the family y = F(x) + C .

Explanation of Solution

Calculation:

The slope field generated by f ( x ) should generate all curves in the family y = F(x) + C .

c.

To determine

To explain: how can use the graph of ${\mathrm{y}}_1=\mathrm{F}(\mathrm{x})\text{}\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{y}}_2={\displaystyle \underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}(\mathrm{t})}\mathrm{d}\mathrm{t}$ to support evaluation of the integral.

Answer to Problem 78E

  ${\mathrm{y}}_2$ evaluates to $\mathrm{F}(\mathrm{x})-\mathrm{F}(0),\text{}\text{so}\text{}{\mathrm{y}}_1=\mathrm{F}(\mathrm{x})\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{y}}_2={\displaystyle \underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}(\mathrm{t})\mathrm{d}\mathrm{t}}$ are the same graph except one is a vertical shift of the other.

Explanation of Solution

Calculation:

  ${\mathrm{y}}_2$ evaluates to $\mathrm{F}(\mathrm{x})-\mathrm{F}(0),\text{}\text{so}\text{}{\mathrm{y}}_1=\mathrm{F}(\mathrm{x})\mathrm{a}\mathrm{n}\mathrm{d}{\mathrm{y}}_2={\displaystyle \underset{0}{\overset{\mathrm{x}}{\int }}\mathrm{f}(\mathrm{t})\mathrm{d}\mathrm{t}}$ are the same graph except one is a vertical shift of the other.

d.

To determine

To explain: how can use a table of values for ${\mathrm{y}}_1-{\mathrm{y}}_2,{\mathrm{y}}_1\text{and}{\mathrm{y}}_2$ defined as in part (c ), to support evaluation of the integral.

Answer to Problem 78E

Use the table of values to evaluate ${\mathrm{y}}_1-{\mathrm{y}}_2$ for several points. The result will be constant for all points if the integration was done correctly.

Explanation of Solution

Calculation:

Use the table of values to evaluate ${\mathrm{y}}_1-{\mathrm{y}}_2$ for several points. The result will be constant for all points if the integration was done correctly.

e.

To determine

To explain: how can use graph of f and NDER of F(x) to support evaluation of the integral.

Answer to Problem 78E

The graph of f and NDER of F(x) should correspond exactly to the graph y = f(x).

Explanation of Solution

Calculation:

The graph of f and NDER of F(x) should correspond exactly to the graph y = f(x).

f.

To determine

To illustrate: parts (a )-(e) for $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\sqrt{{\mathrm{x}}^2+1}}$ .

Answer to Problem 78E

  $\mathrm{F}(\mathrm{x})=\sqrt{{\mathrm{x}}^2+1}+\mathrm{C}$ .

Explanation of Solution

Calculation:

  $\begin{array}{l}\mathrm{u}={\mathrm{x}}^2+1\\ \mathrm{d}\mathrm{u}=2\mathrm{x}\mathrm{d}\mathrm{x}\\ {\displaystyle \int \frac{\mathrm{x}}{\sqrt{{\mathrm{x}}^2+1}}}\mathrm{d}\mathrm{x}=\frac{1}{2}{\displaystyle \int \frac{\mathrm{d}\mathrm{u}}{\sqrt{\mathrm{u}}}}=\sqrt{\mathrm{u}}+\mathrm{C}=\sqrt{{\mathrm{x}}^2+1}+\mathrm{C}\\ \mathrm{F}(\mathrm{x})=\sqrt{{\mathrm{x}}^2+1}+\mathrm{C}.\end{array}$