Somewhere Inside, a Tangent Matches the Secant

You'll learnwhat the Mean Value Theorem promises, the two conditions it needs, and where a broken condition breaks the promise.

Join the two ends of a curve with a straight line. Somewhere in between, a tangent runs parallel to it. That is the Mean Value Theorem, and it comes with two conditions that matter. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · THE SECANT, A PARALLEL TANGENT, AND TWO CONDITIONS · v2

a tangent parallel to the secant, at some c strictly inside (a, b)

Before you start

What you're looking at

A curve on the interval [−2, 2]. The dashed line joins its two ends, and its slope is the average rate of change.

The question

Is there a point inside where the tangent has that same slope, and when are you allowed to say yes?

Step 1 / 12