# Change of y vs dy?

• May 1st 2006, 09:05 AM
Change of y vs dy?
Q: Let y = 4 x^2.
Find the change in y, \Delta y when x= 4 and \Delta x = 0.2
Find the differential dy when x= 4 and dx = 0.2

My solution:

dy/dx = 8x
dy = 8x * dx
dy = (8*4)(.2) = 6.4

But 6.4 is the correct answer only for dy and not change of y, ain't they the same thing?
• May 1st 2006, 02:49 PM
ThePerfectHacker
Quote:

Q: Let y = 4 x^2.
Find the change in y, \Delta y when x= 4 and \Delta x = 0.2
Find the differential dy when x= 4 and dx = 0.2

My solution:

dy/dx = 8x
dy = 8x * dx
dy = (8*4)(.2) = 6.4

But 6.4 is the correct answer only for dy and not change of y, ain't they the same thing?

You know that, $dy=f'(x)dx$
At $x=2$ and $dx=.2$
You have,
$dy=f'(4)(.2)$
That is,
$dy=8\cdot 4\cdot .2=6.4$
As I understand you question about $dy$ and $\Delta y$. Consider a smooth graph with a tangent at a point. You move a certain distance from the point. Two things happen, the change in the function which is $\Delta y$ and the change in the tangent line $dy$, the smaller the change in x or $dx$ the closer $dy$ is to $\Delta x$. Because if you look at the tangent line you see that it is almost the same value as the curve itself for small distances. As if to say, the tangent consideres the curve as a line on an interval.