# Thread: Change of y vs dy?

1. ## 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?

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, $\displaystyle dy=f'(x)dx$
At $\displaystyle x=2$ and $\displaystyle dx=.2$
You have,
$\displaystyle dy=f'(4)(.2)$
That is,
$\displaystyle dy=8\cdot 4\cdot .2=6.4$
As I understand you question about $\displaystyle dy$ and $\displaystyle \Delta y$. 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 $\displaystyle \Delta y$ and the change in the tangent line $\displaystyle dy$, the smaller the change in x or $\displaystyle dx$ the closer $\displaystyle dy$ is to $\displaystyle \Delta x$. 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.