Mathematics for Economists

Approximations[pic 1]

ECON1010

10.2 The Mean Value Theorem

Mean Value Theorem – A theorem connecting the average rate of change of a function to its derivative; a theorem stating that for any given arc between to end points, there is a point at which the tangent to the arc is parallel to the secant through its end points NOTE – The second definition probably makes no sense, but it will once you look at the graphical representation below

Rolle's Theorem – A theorem stating that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them NOTE

– This is just a special case of the mean value theorem (which will be made clear by a diagram lower on this page)

Mean Value Theorem and Rolle's Theorem:

• More specifically (than the worded definitions above), the mean value theorem states the following:

Mean Value Theorem: If f is a continuous, differentiable function between a and b, and a and b

are real number such that a < b, there exists a real number c such that a < c < b and

• '(c) = f (b) − f ( a) b − a

• The graph below will help us show this:

[pic 2]

y = f(x)

[pic 3]

y

0

LTan

B

1. Sec

C

A

• Looking at the diagram above, we can see that, given the chord AB, there is a point C on the main curve such that the tangent at C is parallel to the chord. The chord (LSec) is known as the secant of the curve and the line (LTan) is known as the tangent of the curve. If we denote the x-coordinates of A, B and C by a, b and c respectively, then the slope of the tangent, f '(c), is clearly equal to the slope of the chord, thus the value of c satisfies the mean value theorem equation above

• In the graph above, there happens to be just one real number c with the required property, but in other cases there can be more than one. This possibility occurs when Rolle's theorem is satisfied. Rolle's theorem effectively states that following:

Rolle's Theorem: If f is a continuous, differentiable function between a and b, and a and b are

real number such that f(a) = f(b), there exists a real number c such that a < c < b and