Changes of y Along the Curve

Given a curve y = f (x), suppose that x starts out with the value a and then changes by an infinitesimal amount Δx. What happens to y? Along the curve, y will change by the amount

f(a + Δx) - f(a) = Δy.

But along the tangent line y will change by the amount

l(a + Δx) - l(a) = [f'(a)(a + Δx - a) + b] - [f'(a)(a - a) + b] = f'(a) Δx.

When x changes from a to a + Δx, we see that:

change in y along curve = f(a + Δx) - f(a),

change in y along tangent line = f'(a) Δx.

In the last section we introduced the dependent variable Δy, the increment of y, with the equation

Δy = f (x + Δx) - f(x).

Δy is equal to the change in y along the curve as x changes to x + Δx.

The following theorem gives a simple but useful formula for the increment Δy.