Difference Quotient
This is the "Difference Quotient":
f(x+Δx) − f(x)Δx
It gives the average slope between two points on a curve f(x) that are Δx apart:
Example: find the average slope of f(x) = x2 − 2x + 1
at x = 3 and Δx = 0.1
Evaluate f(x) at x=3:
f(3) = 32 − 2×3 + 1 = 4
Now for f(x+Δx):
f(3.1) = (3.1)2 − 2×3.1 + 1 = 4.41
And the Difference Quotient is:
f(3.1) − f(3)0.1 = 4.41 − 40.1 = 0.410.1 = 4.1
Let's try a smaller value of Δx:
Example continued: try Δx = 0.01
f(3.01) = (3.01)2 − 2×3.01 + 1 = 4.0401
So we have:
f(3.01) − f(3)0.01 = 4.0401 − 40.01 = 0.04010.01 = 4.01
As Δx heads towards 0, the value of the slope heads towards the true slope at that point.
In this case, as Δx gets smaller the slope seems to be heading towards 4, right?
Well, that is the idea behind derivatives, which can find the answer exactly (without guesses) by having Δx head towards 0.