Somewhere Inside, the Tangent Runs Parallel to the Secant

You'll learnto check the Mean Value Theorem on the two different intervals its hypotheses actually live on, and to say what its conclusion promises — at least one c, strictly inside, and a value of f′ rather than of f.

Unit 4 pointed the derivative at motion and at rates. Unit 5 turns it back on the function itself, and this theorem is the hinge — it is what licenses every argument in the unit that reads a fact about f off a fact about f′. Join the two endpoints of an interval with a straight line and measure its slope: that is the average rate across the whole stretch. The theorem says some tangent strictly inside matches it exactly. It comes with two hypotheses, stated on two different intervals, and they are not packaging. Skip them and the theorem still hands back an answer, which is the whole problem.

8 STEPS · 6 QUICK CHECKS · CONTINUOUS ON THE CLOSED, DIFFERENTIABLE ON THE OPEN · v1

f′(c) = (f(b) − f(a)) / (b − a), for at least one c strictly inside (a, b)

Before you start

What you're looking at

One interval, [−2, 2], and a sequence of functions drawn on it. The dashed line joining the two endpoints is the secant, and its slope is the average rate of change across the whole interval.

The question

Is there a point strictly inside where the tangent has that same slope — and what has to be true of f before you are allowed to say yes?

Watch for

Where each failure sits. A derivative that blows up at an endpoint costs nothing; a corner one step inside costs everything.

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