What I can do is to lead you to some pages that explain the difference in
dy and ∆y
I thought this youtube did a good job
http://www.youtube.com/watch?v=xoAnLgs6hiM
There are also numerous other youtube links listed in the right column
btw, I agree with your answers.
Please, don't assume that we've been taught this stuff because we were literally just handed a packet and told to learn it ourselves. I don't understand this in the slightest.
The question is: Use the information to evaluate and compare delta y and dy. y=2-x^4, x=2 and delta x = dy= 0.01
What I've done so far is differentiate, to get that dy=-4x^3. In the packet, it looks like they plugged 2 in for x and multiplied by dx, so I did, and I got -.32
For delta y, I set up the equation like f(2+0.01) - f(2), and I got -.3224
Are these right? Is there some way that I'm supposed to compare them using words? If I don't even understand what I'm doing, then how am I supposed to do so?
2 answers
Delta y is the Actual change of y when x changes from x to x+delta x. (I capitalized Actual so that you'd see the top part of A looks like delta - a way to remember what delta y is.) dy is the estimated change in y when x changes from x to x+delta x when you use the tangent line at x to estimate.
To help you see this, let's look at the Actual change of y, or delta y.
x starts at 2 and moves 0.01 to x=2.01
f(2)=2-2^4=-14
f(2.01)=2-2.01^4=-14.32240801
Therefore the change of y, delta y = -0.32240801
The equation of the tangent line at x=2 is y=-32x+50. (This requires work on the "side.")
If x=2, y=-64+50+-14
If x=2.01, y = -64.32+50=-14.32
Therefore dy=-.32
But that's too much work!
Remember calc notation:
dy/dx=f'(x)
"Multiplying" that equation by dx gives
dy=f'(x)*dx
This is a faster way to find dy!
f(x)=2-x^4
f'(x)=-4x^3
At x=2,
dy=(-4*2^3)*(.01)=-.32
Notice this is what we got with the tangent line approximation which is very close to the "Actual" change of y or delta y using the "Actual" function!
LON
To help you see this, let's look at the Actual change of y, or delta y.
x starts at 2 and moves 0.01 to x=2.01
f(2)=2-2^4=-14
f(2.01)=2-2.01^4=-14.32240801
Therefore the change of y, delta y = -0.32240801
The equation of the tangent line at x=2 is y=-32x+50. (This requires work on the "side.")
If x=2, y=-64+50+-14
If x=2.01, y = -64.32+50=-14.32
Therefore dy=-.32
But that's too much work!
Remember calc notation:
dy/dx=f'(x)
"Multiplying" that equation by dx gives
dy=f'(x)*dx
This is a faster way to find dy!
f(x)=2-x^4
f'(x)=-4x^3
At x=2,
dy=(-4*2^3)*(.01)=-.32
Notice this is what we got with the tangent line approximation which is very close to the "Actual" change of y or delta y using the "Actual" function!
LON
2 answers