Average slope and tangent slope
The average slope of a curve from A to B is how much it rises divided by how far it goes sideways. The straight line through A and B is called the secant.
The tangent slope at one point is the slope of the line that just touches the curve there. That is the derivative f'(x).
Average slope = (f(b) − f(a)) ÷ (b − a). It is one number for the whole piece. The tangent slope changes from point to point.
The mean value theorem
Suppose a function f is
- continuous on the closed interval [a, b] (you can draw it without lifting the pencil), and
- differentiable on the open interval (a, b) (no sharp corners or breaks inside).
Then there is at least one point c between a and b such that
f'(c) = (f(b) − f(a)) / (b − a)
In words: somewhere on the curve the tangent is parallel to the secant AB. Slide the red line in the 3D and you will find it.
Rolle's theorem: the flat special case
If the two ends of the curve are at the same height, f(a) = f(b), then the average slope is 0. So the theorem says there is a point c where f'(c) = 0: the tangent is flat. This is a hill-top or a valley-bottom.
Rolle's theorem needs three things: continuous on [a, b], differentiable on (a, b), and f(a) = f(b).
Why the conditions matter
If the curve has a corner inside, the tangent is not defined there and the theorem can fail. Example: f(x) = |x| on [−1, 2]. The average slope is (2 − 1) ÷ 3 = 1/3, but the slope is only −1 or +1 and never 1/3, because the corner at 0 breaks differentiability.
If the curve has a gap (a jump), it is not continuous, and the tangent can never match the secant. Always check the conditions before using the theorem.
How to find c in a problem
- Check that f is continuous on [a, b] and differentiable on (a, b).
- Work out the average slope (f(b) − f(a)) ÷ (b − a).
- Find f'(x) and solve f'(c) = average slope.
- Keep only the answers that lie inside (a, b).
The theorem is also used to prove that a function with f'(x) = 0 everywhere is constant, and to find bounds on how much a function can change.
Try it yourself
In the 3D: before you move the slider, guess where the green spot will be. Then slide c and check. Next move B so the curve is higher or lower, and watch the green spot move.
At home: walk 10 steps along a path, timing yourself. Your average speed is 10 steps ÷ time. Ask a friend to watch you: was there a moment you moved exactly at that speed? The theorem says yes, as long as your speed changes smoothly.
Key formulas and definitions
- Average slope = (f(b) − f(a)) / (b − a)
- Mean value theorem: f'(c) = (f(b) − f(a)) / (b − a), for some c in (a, b)
- Rolle's theorem: if f(a) = f(b), then f'(c) = 0 for some c in (a, b)
- Conditions: continuous on [a, b], differentiable on (a, b)
Worked examples
1. Find c for f(x) = x² on [1, 3].
Step 1: f is a polynomial, so it is smooth. Step 2: average slope = (9 − 1) / (3 − 1) = 4. Step 3: f'(x) = 2x, so 2c = 4 and c = 2. Step 4: 2 lies in (1, 3), so c = 2.
2. Check Rolle's theorem for f(x) = x² − 4x + 3 on [0, 4] and find c.
f(0) = 3 and f(4) = 16 − 16 + 3 = 3, so f(0) = f(4). It is a polynomial, so the other conditions hold. f'(x) = 2x − 4. Set 2c − 4 = 0, so c = 2, which is inside (0, 4).
3. Find c for f(x) = x³ on [0, 2].
Average slope = (8 − 0) / 2 = 4. f'(x) = 3x², so 3c² = 4 and c = ±2/√3 ≈ ±1.155. Only +1.155 lies in (0, 2). So c ≈ 1.155.
4. A car covers 150 km in 2 hours. What does the mean value theorem say about its speed?
Average speed = 150 / 2 = 75 km/h. If the position changes smoothly, at some moment the speedometer shows exactly 75 km/h.
5. Find c for f(x) = ln x on [1, e].
Average slope = (ln e − ln 1) / (e − 1) = 1 / (e − 1). f'(x) = 1/x, so 1/c = 1/(e − 1) and c = e − 1 ≈ 1.718. This is inside (1, e).
6. Does the mean value theorem work for f(x) = |x| on [−1, 2]?
No. f is not differentiable at x = 0, which lies inside the interval. The average slope is 1/3 but f'(x) is only −1 or 1, so no c exists. The condition fails.
Common mistakes
- Forgetting to check the conditions. A corner or a gap inside the interval can make the theorem fail.
- Giving a value of c that lies outside (a, b). Always check that c is between the end points.
- Mixing up the two theorems. Rolle's needs f(a) = f(b); the mean value theorem does not.
- Thinking c is the middle point. c is just where the tangent slope matches; it is usually not the midpoint.