Question

slope fields and eulers

1. let \\(\frac{dy}{dx} = \frac{-x}{y}\\).
2. for the ordered pairs shown in the table below, compute the value of the derivative.

x	y	dy/dx	x	y	dy/dx
-3	3		3	-3
0	3		3	0
2	1		1	2
-1	-1		1	1
2	2		-2	2
-2	-2		2	0
0	2		2	3
3	2		-2	-3
-3	-2		3	-2
2	-3		1	-2
2	-1		-2	1
-2	0		0	-1
0	-2		-3	1
-1	2		0	1

Explanation:

Step1: Substitute x,y into $\frac{dy}{dx}=\frac{-x}{y}$

For each ordered pair $(x,y)$, calculate $\frac{dy}{dx}=\frac{-x}{y}$ (note: $y
eq0$; if $y=0$, derivative is undefined).

Left Column Pairs:

1. $x=-3,y=3$: $\frac{dy}{dx}=\frac{-(-3)}{3}=1$
2. $x=0,y=3$: $\frac{dy}{dx}=\frac{-0}{3}=0$
3. $x=2,y=1$: $\frac{dy}{dx}=\frac{-2}{1}=-2$
4. $x=-1,y=-1$: $\frac{dy}{dx}=\frac{-(-1)}{-1}=-1$
5. $x=2,y=2$: $\frac{dy}{dx}=\frac{-2}{2}=-1$
6. $x=-2,y=-2$: $\frac{dy}{dx}=\frac{-(-2)}{-2}=-1$
7. $x=0,y=2$: $\frac{dy}{dx}=\frac{-0}{2}=0$
8. $x=3,y=2$: $\frac{dy}{dx}=\frac{-3}{2}$
9. $x=-3,y=-2$: $\frac{dy}{dx}=\frac{-(-3)}{-2}=-\frac{3}{2}$
10. $x=2,y=-3$: $\frac{dy}{dx}=\frac{-2}{-3}=\frac{2}{3}$
11. $x=2,y=-1$: $\frac{dy}{dx}=\frac{-2}{-1}=2$
12. $x=-2,y=0$: $\frac{dy}{dx}$ is undefined
13. $x=0,y=-2$: $\frac{dy}{dx}=\frac{-0}{-2}=0$
14. $x=-1,y=2$: $\frac{dy}{dx}=\frac{-(-1)}{2}=\frac{1}{2}$

Right Column Pairs:

1. $x=3,y=-3$: $\frac{dy}{dx}=\frac{-3}{-3}=1$
2. $x=3,y=0$: $\frac{dy}{dx}$ is undefined
3. $x=1,y=2$: $\frac{dy}{dx}=\frac{-1}{2}$
4. $x=1,y=1$: $\frac{dy}{dx}=\frac{-1}{1}=-1$
5. $x=-2,y=2$: $\frac{dy}{dx}=\frac{-(-2)}{2}=1$
6. $x=2,y=0$: $\frac{dy}{dx}$ is undefined
7. $x=2,y=3$: $\frac{dy}{dx}=\frac{-2}{3}$
8. $x=-2,y=-3$: $\frac{dy}{dx}=\frac{-(-2)}{-3}=-\frac{2}{3}$
9. $x=3,y=-2$: $\frac{dy}{dx}=\frac{-3}{-2}=\frac{3}{2}$
10. $x=1,y=-2$: $\frac{dy}{dx}=\frac{-1}{-2}=\frac{1}{2}$
11. $x=-2,y=1$: $\frac{dy}{dx}=\frac{-(-2)}{1}=2$
12. $x=0,y=-1$: $\frac{dy}{dx}=\frac{-0}{-1}=0$
13. $x=-3,y=1$: $\frac{dy}{dx}=\frac{-(-3)}{1}=3$
14. $x=0,y=1$: $\frac{dy}{dx}=\frac{-0}{1}=0$

Answer:

$x$	$y$	$dy/dx$	$x$	$y$	$dy/dx$
$0$	$3$	$0$	$3$	$0$	Undefined
$2$	$1$	$-2$	$1$	$2$	$-\frac{1}{2}$
$-1$	$-1$	$-1$	$1$	$1$	$-1$
$2$	$2$	$-1$	$-2$	$2$	$1$
$-2$	$-2$	$-1$	$2$	$0$	Undefined
$0$	$2$	$0$	$2$	$3$	$-\frac{2}{3}$
$3$	$2$	$-\frac{3}{2}$	$-2$	$-3$	$-\frac{2}{3}$
$-3$	$-2$	$-\frac{3}{2}$	$3$	$-2$	$\frac{3}{2}$
$2$	$-3$	$\frac{2}{3}$	$1$	$-2$	$\frac{1}{2}$
$2$	$-1$	$2$	$-2$	$1$	$2$
$-2$	$0$	Undefined	$0$	$-1$	$0$
$0$	$-2$	$0$	$-3$	$1$	$3$
$-1$	$2$	$\frac{1}{2}$	$0$	$1$	$0$