The Mean Value Theorem
Over an interval where a function is continuous and differentiable, some interior point has instantaneous rate equal to the average rate across the whole interval. Most facts linking a derivative to a function's behaviour rest on this.
What a learner can do afterwards
- State the hypotheses and find the guaranteed point for a given function
- Give a function that fails the theorem and identify the hypothesis it breaks
- Use the theorem to argue that a zero derivative on an interval forces a constant function
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Take a car that covers 90 km in one hour. Its average speed is 90 km per hour, even if it sped up and slowed down along the way. The mean value theorem says the speedometer showed exactly 90 at some instant. In symbols, the average rate from a to b is f of b minus f of a, over b minus a. The theorem promises a point c inside where f prime of c equals that average.
The promise needs two conditions. The function must be continuous on the closed interval from a to b, and differentiable on the open interval between them. Endpoints only need continuity, since a derivative wants slopes from both sides. Picture the secant line joining the endpoints: the theorem gives a tangent line parallel to it, touching the curve somewhere inside. When the endpoint values match, the secant is flat, so some c has f prime of c equals 0. For g of x equals x squared minus 4 x plus 3 on 1 to 3, both values are 0 and c equals 2.
Take f of x equals 2 x squared on 1 to 3. First the average rate: f of 1 is 2 and f of 3 is 18, so 18 minus 2 over 3 minus 1 is 8. The derivative is 4 x, so set 4 c equals 8 and get c equals 2, which sits inside. A polynomial passes everywhere, so both conditions hold with no extra work. A second example: f of x equals x minus 3 over x on 1 to 3. The average rate is 2, and f prime of x is 1 plus 3 over x squared. Setting that equal to 2 gives c squared equals 3, so c is root 3, about 1.73, inside the interval.
When a condition fails, the promise evaporates. The absolute value of x minus 2 has a sharp corner at x equals 2, where the slope jumps from negative 1 to 1, so no derivative exists there. And 1 over x blows up at x equals 0, so it is not continuous on any closed interval holding 0. Failing the conditions means no guarantee, though the conclusion may still hold by luck.
The theorem also bounds growth and proves constancy. If f of 1 is 2 and the derivative stays at most 5, then f of 4 is at most 2 plus 3 times 5, which is 17. And if the derivative is 0 at every interior point, then f of b minus f of a equals 0 times b minus a, so every value matches and the function is constant.
A smooth curve always has an interior point whose tangent matches its secant.
More to read
Mean value theorem review
The mean value theorem connects a function's average rate of change to its derivative. If a function f is differentiable on an interval from a to b, then somewhere inside that interval there is a number c where f'(c) equals the average rate of change (f(b) - f(a)) / (b - a). Picture the secant line joining the two endpoints of the curve. The theorem promises that the curve has at least one tangent line parallel to that secant line, touching the graph at a point in between. To use the theorem well, always say its two hypotheses out loud first: the function must be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Once those hold, the guaranteed point c exists, even if the theorem does not tell you exactly where it is.
Establishing differentiability for MVT
The mean value theorem is an existence theorem, a cousin of the intermediate value theorem and the extreme value theorem. It promises that a certain point exists without handing it to you. Before you can claim anything from it, you must check its conditions. The function needs to be continuous over the whole closed interval from a to b, and differentiable over the open interval between them. Why the split? Endpoints can behave badly for derivatives, since a derivative wants slopes from both sides, so the theorem only asks continuity there. A polynomial like x^3 - x passes everywhere. But watch for functions like 1/x, which breaks at 0, or |x|, which has a sharp corner. If a condition fails on your interval, the theorem simply gives no guarantee, and a counterexample may be hiding nearby.
Average rate of change review
The average rate of change of a function f over the interval from a to b is (f(b) - f(a)) / (b - a). Read it as: how much did the output change, divided by how much the input changed. It measures how fast the function grew or shrank per unit, on average, across that stretch. Geometrically, it is the slope of the straight line connecting the two endpoints of the graph. If a car goes from mile marker 10 to mile marker 70 in one hour, its average speed is 60 miles per hour, even if it sped up and slowed up along the way. The mean value theorem says that for a smooth trip, there was at least one instant when the speedometer showed exactly that average. Mastering this formula first makes the theorem feel natural instead of mysterious.
Intermediate value theorem review
The intermediate value theorem describes what continuous functions do. If f is continuous on [a, b], then between x = a and x = b the function takes on every value between f(a) and f(b). Formally, for any number L between f(a) and f(b), there is a c in [a, b] with f(c) = L. The graph cannot skip a height, because a continuous curve has no gaps. This matters here because the mean value theorem works the same way: both are existence theorems that guarantee a point with a special property, as long as continuity holds. IVT promises a point with a target output. MVT promises a point with a target slope. Keeping the two side by side in your mind makes each one easier to remember and to state carefully.
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Where it sits
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.