Mean value theorem

For a function continuous on a closed interval and differentiable inside it, some interior derivative equals the average rate of change.

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Mean Value Theorem
The Organic Chemistry Tutor · English · Explanation
Connection evidence
For $a<b$ , continuity on [a,b] and differentiability on (a,b) guarantee an interior point where the tangent slope equals the secant slope. For $f(x)=x$ ²− $4x+1$ on [1,5], the secant slope is ( $6-(-2)$ )/4=2. Solving $2c-4=2$ gives $c=3$ , inside the interval. The cusp of $x^(2/3)$ at the endpoint 0 does not violate differentiability on (0,1). By contrast | $4x-5$ | has a corner at $5/4$ inside (0,2), so the theorem does not apply there.
Mean value theorem | Existence theorems | AP Calculus AB | Khan Academy
Khan Academy · English · Explanation
Connection evidence
This 180-second whiteboard clip introduces the Mean Value Theorem by first writing its two hypotheses—continuity on `[a,b]` and differentiability on `(a,b)`—and then translating them into a graph. The speaker explains bracket notation, draws a smooth curve with endpoint values ` $f(a)$ ` and ` $f(b)$ `, states verbally that some interior instantaneous rate of change equals the average rate of change over the interval, and begins the geometric interpretation by identifying the average change with the slope of the secant line. The clip stops before the full algebraic formula and tangent-line comparison are shown. This video segment provides a visual and intuitive explanation of the Mean Value Theorem in calculus. Using a whiteboard format, the instructor draws a function curve over an interval [a, b] and illustrates the secant line connecting the endpoints. The core concept is demonstrated by drawing tangent lines at specific points that are parallel to the secant line, showing where the instantaneous rate of change equals the average rate of change. The instructor then formally writes out the theorem's hypotheses (continuity on [a, b], differentiability on (a, b)) and its mathematical conclusion: there exists a point c in (a, b) such that f'(c) = ( $f(b) - f(a)$ ) / (b - a). Geometric representations of the change in y and change in x are used to build the average rate of change formula step-by-step. This 36-second Khan Academy whiteboard clip reviews the mean value theorem by pairing the formal statement with a graph. The board states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then there exists some $c\in(a,b)$ with $\frac{\Delta y}{\Delta x}=\frac{f(b)-f(a)}{b-a}=f'(c)$ . The speaker unpacks the notation in words, explaining that somewhere inside the interval the instantaneous rate of change equals the average rate of change over the whole interval. The diagram shows the curve $y=f(x)$ , the secant line through the endpoints, and a parallel tangent-like line at an interior point, visually encoding the equality of slopes. The clip also emphasizes that the theorem guarantees existence rather than uniqueness, since the speaker notes that more than one point could serve as $c$ . No numerical example or rigorous proof is given in this excerpt; instead, it functions as a conceptual bridge from symbolic hypotheses to geometric intuition.
The MEAN Value Theorem is Actually Very Nice
Dr. Trefor Bazett · English · Explanation
Connection evidence
The secant joins the endpoint values at $a$ and $b$ . The theorem guarantees a tangent with the same slope at some interior point when its hypotheses hold. For $a<b$ , if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , there exists $c\in(a,b)$ such that $f'(c)=\frac{f(b)-f(a)}{b-a}$ . The point need not be unique. A sharp interior corner or a jump at an endpoint can remove the guarantee. The examples explain why the theorem needs these hypotheses as a sufficient framework; they do not assert that its conclusion is impossible for every function outside that framework. Under the same continuity and differentiability assumptions, if $f(a)=f(b)$ , the secant slope is zero. Therefore some interior point has $f'(c)=0$ , corresponding to a horizontal tangent.

Questions and answers about Mean value theorem

Why do the continuity and differentiability hypotheses of the mean value theorem matter?

The hypotheses ensure that the function behaves smoothly enough for an interior tangent to match the secant slope. A sharp interior corner or a jump at an endpoint can remove this guarantee. These conditions provide a sufficient framework; without them, the conclusion may fail because the derivative might not exist at the required point or the function might not be defined continuously across the interval.

The function fails to be continuous on $[a,b]$ or differentiable on $(a,b)$ .

Example: Graphical counterexamples show that if a function has a sharp corner or a jump, there may be no point $c$ where the tangent is parallel to the secant line.

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Is the point c guaranteed by the mean value theorem unique?

No, the point $c$ is not necessarily unique. The theorem guarantees the existence of at least one such point in the open interval $(a,b)$ , but there may be multiple points where the instantaneous rate of change equals the average rate of change.

$f$ is continuous on $[a,b]$
$f$ is differentiable on $(a,b)$

Example: For a sinusoidal function over a full period or half-period, the tangent slope may match the secant slope at more than one point.

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What does the mean value theorem state for a function continuous on [a,b] and differentiable on (a,b)?

The mean value theorem states that if a function $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b)$ , then there exists at least one point $c$ in $(a,b)$ such that the instantaneous rate of change (derivative) at $c$ equals the average rate of change over the interval. Mathematically, this is expressed as $f'(c) = \frac{f(b)-f(a)}{b-a}$ .

$a < b$
$f$ is continuous on $[a,b]$
$f$ is differentiable on $(a,b)$

Example: For a function satisfying the hypotheses, the theorem guarantees a tangent line at some interior point $c$ that is parallel to the secant line connecting the endpoints $(a, f(a))$ and $(b, f(b))$ .

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What is the geometric interpretation of the mean value theorem regarding secants and tangents?

Geometrically, the mean value theorem states that if a function is smooth over an interval, there is at least one point inside the interval where the tangent line is parallel to the secant line connecting the endpoints of the interval. The secant represents the average rate of change, and the tangent represents the instantaneous rate of change.

$f$ is continuous on $[a,b]$
$f$ is differentiable on $(a,b)$

Example: The theorem guarantees a tangent with the same slope at some interior point when its hypotheses hold. The secant joins the endpoint values at $a$ and $b$ .

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How does Rolle's theorem relate to the mean value theorem?

Rolle's theorem is a special case of the mean value theorem. It applies when the function values at the endpoints are equal, i.e., $f(a) = f(b)$ . In this scenario, the average rate of change $\frac{f(b)-f(a)}{b-a}$ becomes zero. Consequently, the mean value theorem guarantees the existence of a point $c$ in $(a,b)$ where the derivative $f'(c)$ is zero, corresponding to a horizontal tangent.

$f$ is continuous on $[a,b]$
$f$ is differentiable on $(a,b)$
$f(a) = f(b)$

Example: If a function starts and ends at the same height on an interval, Rolle's theorem ensures there is at least one peak or valley (horizontal tangent) in between.

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More concepts to explore

Continuity
Continuity at a point means that the function value equals its limit there; continuity on an interval requires this at each relevant point.
Derivatives
A derivative is the limit of a difference quotient, measuring local rate of change when that limit exists.
Antiderivatives
An antiderivative has the given function as its derivative; on an interval, any two antiderivatives differ by a constant.
Chain rule
The derivative of a differentiable composition combines the derivative of the outer function with that of the inner function.
Change of variables
A suitable coordinate change transforms volume elements through the absolute Jacobian determinant.

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Continuity · Prerequisite · Map context
Derivatives · Prerequisite · Map context

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