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Created page with "==Statement== Suppose <math>f</math> is a function defined on a closed interval <math>[a,b]</math> (with <math>a < b</math>) such that the following two conditions hold: # ..."

==Statement==

Suppose <math>f</math> is a function defined on a [[closed interval]] <math>[a,b]</math> (with <math>a < b</math>) such that the following two conditions hold:

# <math>f</math> is a [[continuous function]] on the [[closed interval]] <math>[a,b]</math> (i.e., it is right continuous at <math>a</math>, left continuous at <math>b</math>, and two-sided continuous at all points in the open interval <math>(a,b)</math>).

# <math>f</math> is a [[differentiable function]] on the [[open interval]] <math>(a,b)</math>, i.e., the derivative exists at all points in <math>(a,b)</math>. Note that we ''do not'' require the [[derivative]] of <math>f</math> to be a continuous function.

Then, there exists <math>c</math> in the open interval <math>(a,b)</math> such that the derivative of <math>f</math> at <math>c</math> equals the [[difference quotient]] <math>\Delta f(a,b)</math>. More explicitly:

<math>f'(c) = \frac{f(b) - f(a)}{b - a}</math>

Geometrically, this is equivalent to stating that the [[tangent line]] to the [[graph]] of <math>f</math> at <math>c</math> is parallel to the [[chord]] joining the points <math>(a,f(a))</math> and <math>(b,f(b))</math>.

Note that the theorem simply guarantees the existence of <math>c</math>, and does not give a formula for finding such a <math>c</math> (which may or may not be unique).

==Related facts==

* [[Rolle's theorem]]

* [[Zero derivative implies locally constant]]

* [[Fundamental theorem of calculus]]

* [[Positive derivative implies increasing]]

* [[Increasing and differentiable implies nonnegative derivative]]

Suppose <math>f</math> is a function defined on a [[closed interval]] <math>[a,b]</math> (with <math>a < b</math>) such that the following two conditions hold:

# <math>f</math> is a [[continuous function]] on the [[closed interval]] <math>[a,b]</math> (i.e., it is right continuous at <math>a</math>, left continuous at <math>b</math>, and two-sided continuous at all points in the open interval <math>(a,b)</math>).

# <math>f</math> is a [[differentiable function]] on the [[open interval]] <math>(a,b)</math>, i.e., the derivative exists at all points in <math>(a,b)</math>. Note that we ''do not'' require the [[derivative]] of <math>f</math> to be a continuous function.

Then, there exists <math>c</math> in the open interval <math>(a,b)</math> such that the derivative of <math>f</math> at <math>c</math> equals the [[difference quotient]] <math>\Delta f(a,b)</math>. More explicitly:

<math>f'(c) = \frac{f(b) - f(a)}{b - a}</math>

Geometrically, this is equivalent to stating that the [[tangent line]] to the [[graph]] of <math>f</math> at <math>c</math> is parallel to the [[chord]] joining the points <math>(a,f(a))</math> and <math>(b,f(b))</math>.

Note that the theorem simply guarantees the existence of <math>c</math>, and does not give a formula for finding such a <math>c</math> (which may or may not be unique).

==Related facts==

* [[Rolle's theorem]]

* [[Zero derivative implies locally constant]]

* [[Fundamental theorem of calculus]]

* [[Positive derivative implies increasing]]

* [[Increasing and differentiable implies nonnegative derivative]]