Let the Point Go and the Slope Becomes a Function

You'll learnhow a chord between two points turns into the tangent at one, and why leaving the point unnamed gives you a whole new function.

Pick a point on a curve and a second point a little to its right. The line through them has a slope you can compute. Now slide the second point in and watch where that slope goes. Then stop naming the point at all, and see what comes out. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · h → 0, NEVER x → 0 · f′(x) IS A FUNCTION, f′(3) IS ONE NUMBER · v2

f(x) = x² − 4x · f′(x) = 2x − 4 · f′(3) = 2

Before you start

What you're looking at

The curve f(x) = x² − 4x, with one point marked on it at x = 3.

The question

How steep is the curve right at that point, and what happens if we never name the point at all?

Step 1 / 12