Calculus I · The Mean Value Theorem

Where this is heading

You now hold the proof engine for the back half of Book 2. Chapter 7 (Taylor) uses the MVT — in its Cauchy form — to control the error when you approximate a curve by a polynomial, turning “the tangent line is close” into “and here is exactly how close.” Chapter 8 (maxima, minima, convexity) runs almost entirely on Consequence 2: you’ll find a function’s highest and lowest points by tracking where its derivative changes sign, you’ll read its curvature off the second derivative, and you’ll meet convex functions — the ones that stay above all their tangent lines — which are precisely the functions on which the gradient descent of Chapter 9 is guaranteed to find the true bottom. Every one of those results is a consequence of the theorem you proved today.

Continuity forced structure. Differentiability forces more. That is the whole secret of calculus, and the Mean Value Theorem is where you first see it with full force.

Problems

Six tiers. Fluency and Practice are doing. Concept & counterexample and ◆ Diamond are thinking* — and in this chapter the Diamond tier is the heart of it, because the MVT’s value is in its proofs. Teach it is teaching. Looking ahead reaches into Chapters 7 and 8. Do all of Fluency; the Concept and Diamond tiers are where this chapter is actually learned.*

Fluency — do these fast

1. Verify Rolle’s hypotheses hold, then find every $c$ it guarantees: (a) $f(x) = x^2 - 4x$ on $[0,4]$; (b) $f(x) = x^3 - x$ on $[-1,1]$.
2. Find every $c$ the MVT guarantees: (a) $f(x) = x^2$ on $[1,3]$; (b) $f(x) = x^3$ on $[0,3]$.
3. For each, name which single hypothesis fails, so the theorems don’t apply: (a) $f(x) = |x|$ on $[-1,1]$; (b) $f(x) = \tfrac1x$ on $[-1,1]$; (c) $f(x) = \lfloor x\rfloor$ (the step function) on $[0,2]$.
4. Given $f'(x) = (x-1)(x-3)$, state the intervals where $f$ is increasing and where it is decreasing.
5. A car’s position is $s(t) = t^2$ miles after $t$ hours, on $[0,3]$. Find its average speed over the trip, and the exact instant its speedometer matched that average.
6. If $|f'(x)| \le 3$ for all $x$ and $f(0) = 5$, what is the largest $f(4)$ could possibly be? The smallest?

Practice — multi-step

1. Find all $c$ guaranteed by the MVT for $f(x) = x^3 - x$ on $[0,2]$.
2. Show $f(x) = x^5 + 2x + 1$ has exactly one real root. (Existence: use the Intermediate Value Theorem on a sign change. Uniqueness: compute $f'$ and use Consequence 2.)
3. Two functions satisfy $f'(x) = g'(x) = 2x$ for all $x$, with $f(0) = 3$ and $g(0) = 5$. Find $f(x) - g(x)$, then $f(x)$ and $g(x)$ themselves, citing the consequence you use.
4. Prove $|e^{-a} - e^{-b}| \le |a - b|$ for all $a, b \ge 0$. (Apply the MVT to $f(x) = e^{-x}$ and bound $|f'|$.)
5. Prove the inequality $x > \ln(1+x)$ for all $x > 0$. (Let $h(x) = x - \ln(1+x)$, show $h(0) = 0$ and $h'(x) > 0$ for $x > 0$, then use Consequence 2.)
6. A differentiable $f$ has $f(1) = 3$ and $f'(x) \ge 2$ for all $x$ in $[1,5]$. Prove $f(5) \ge 11$. (A one-sided version of Consequence 3.)

Concept & counterexample — think; true/false with justification, or build the example

1. 1. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, some interior $c$ has $f'(c)$ equal to the average slope. (b) If $f'(c) = 0$ at some interior $c$, then $f(a) = f(b)$. (c) If $f' > 0$ on $(a,b)$, then $f$ is increasing there. (d) The MVT tells you how to find the point $c$.
2. Find the error. A student applies Rolle to $f(x) = |x|$ on $[-1,1]$: “since $f(-1) = f(1) = 1$, there must be a $c$ with $f'(c) = 0$.” But $|x|$ has slope $\pm1$ everywhere it has a slope at all. What went wrong?
3. Construct a function on $[0,1]$ that is continuous with $f(0) = f(1)$ but has $f'(c) = 0$ at no interior point. (You’ll have to break one hypothesis — which one, and why is that allowed to ruin the conclusion?)
4. Construct a function that is increasing on an interval yet has $f'= 0$ at one point inside it. (This shows “$f$ increasing” does not require “$f' > 0$ everywhere” — the converse of Consequence 2 is false.)
5. Is it possible? A differentiable function with $f(a) = f(b)$ but $f'(x) \ne 0$ at every point of $(a,b)$. Argue from Rolle.
6. Explain in two or three sentences why “$f' = g'$ everywhere” does not mean “$f = g$,” and state exactly what it does mean, with a one-line example.

◆ Diamond — the proofs are the chapter

1. ◆ Prove Fermat’s lemma: if $f$ has an interior local maximum at $c$ and $f'(c)$ exists, then $f'(c) = 0$. (Use one-sided difference quotients; reproduce the text’s sign argument.)
2. ◆ Prove Rolle’s theorem from the Extreme Value Theorem and Fermat’s lemma, handling the case where both extremes sit at the endpoints.
3. ◆ Prove the Mean Value Theorem from Rolle by the tilt trick: construct the auxiliary $g$, verify it satisfies Rolle’s hypotheses, and finish.
4. ◆ Prove Consequence 1 ($f' = 0 \Rightarrow f$ constant) and its corollary ($f' = g' \Rightarrow f - g$ constant). State clearly where you invoke the MVT.
5. ◆ Prove Consequence 2 ($f' > 0$ on an interval $\Rightarrow f$ strictly increasing).
6. ◆◆ Prove Cauchy’s generalized MVT by applying Rolle to $h(x) = [f(b)-f(a)]g(x) - [g(b)-g(a)]f(x)$. Verify $h(a) = h(b)$ first.
7. ◆◆ Use the MVT (or a monotonicity argument) to prove $e^x \ge 1 + x$ for all real $x$, with equality only at $x = 0$. (Let $h(x) = e^x - 1 - x$; analyze the sign of $h'$ on each side of $0$.) Note for later: $1 + x$ is the tangent line to $e^x$ at $0$, and you’ve just shown $e^x$ never dips below its tangent.

Teach it — explain it, write it up, or fix it

1. In three or four sentences with the driving/speedometer story, teach a friend what the MVT guarantees — and be careful to say what it does not give you (it doesn’t tell you where $c$ is).
2. Debug your classmate. They say: “$f(x) = x^4 - 2x^2$ has $f(-\sqrt2) = f(\sqrt2) = 0$, so by Rolle there is exactly one point in between with zero slope.” Pinpoint the precise misreading and state what Rolle actually promises. (How many such points does this $f$ really have?)
3. Write a complete, textbook-quality worked solution proving “$x^5 + 2x + 1$ has exactly one real root,” narrating the two halves (IVT for existence, monotonicity for uniqueness) so a peer learns the reusable pattern.
4. Design a problem. Invent a cubic $f$ and an interval $[a,b]$ on which Rolle applies, then write the solution key that finds every guaranteed $c$.
5. In two or three sentences, teach the difference between the IVT and the MVT — one is a promise about the values a continuous function must take, the other about the slopes a differentiable function must achieve.

Looking ahead — Chapters 7 and 8 in your hands early

1. The MVT says $f(b) = f(a) + f'(c)(b-a)$ for some midpoint $c$. Chapter 7 sharpens this to $f(b) = f(a) + f'(a)(b-a) + \tfrac12 f''(c)(b-a)^2$ — same shape, one derivative deeper. Guess the next version, with one more derivative still. (What power of $(b-a)$ and what factorial would you expect?)
2. Consequence 2 said the sign of $f'$ controls rising and falling. In Chapter 8 you’ll find maxima and minima by locating where $f'$ changes sign. Predict: at a point where $f$ switches from increasing to decreasing, what is $f'$ doing there, and what kind of point have you found?
3. In problem 25 you proved $e^x$ stays above its tangent line $1+x$ everywhere. A function that lies above all of its tangent lines is called convex — the star of Chapter 8, and the exact condition under which gradient descent (Chapter 9) is guaranteed to reach the true minimum. Predict what the second derivative $f''$ (the slope of the slope) must be doing, in sign, for a convex function.

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