Chapter 7.1, Problem 45E

To determine

To match the slope $\frac{dy}{dx}=\left|x\right|$ with the correct graph and to find the particular solution for the point $\left(0,2\right)$ .

Answer to Problem 45E

The correct graph of the given slope $\frac{dy}{dx}=\left|x\right|$ is,

  

The graph of the point $\left(0,2\right)$ on the given slope is,

  

Explanation of Solution

Given information :

The slope $\frac{dy}{dx}=\left|x\right|$ .

Calculation:

Consider, the slope $\frac{dy}{dx}=\left|x\right|$ .

It is clear that the slope depends only on the value of x so each column of line segments must be the same. Also, the slope is positive at each point of xsince $\frac{dy}{dx}=\left|x\right|$ .

(i.e.,) when $x=3\Rightarrow \frac{dy}{dx}=\left|3\right|=3$ and when $x=-3\Rightarrow \frac{dy}{dx}=\left|-3\right|=3$

Thus, the graph of the slope field is in the form,

  

Therefore, from the given set of graphs option B is the correct match for the slope $\frac{dy}{dx}=\left|x\right|$

Now, to find the particular solution through the above graph for the point $\left(0,2\right)$ .

Consider the point $\left(0,2\right)$ by using the line segment at the point $x=1$ sketch the additional portion of the curve to have the slope and similarly using the line segment at the point $x=0$ again sketch the curve continuing the similar process to fill the entire sketch then the final curve results as,

  

Which is the particular solution for the slope $\frac{dy}{dx}=\left|x\right|$ at the point $\left(0,2\right)$ .