Holiday Blog # 2

Mean Value Theorem

1. Check if the function is continuous.

2. Check for differentiability or non-differentiability.

3. Slope

f ' (c) = f(b) - f(a) / b - a

* It means on an interval [a,b]. There must be some x-value where the derivative = the slope between a & b.

slope of a secant line = slope of a tangent line between a & b

Example:

Given f(x) = 5 - 4/x, find all values of c in the open interval (1,4) such that

f ' (c) = f(4) - f(1) / 4-1

yes it is both continuous and differentiable

= 4-1/4-1 = 1

5 - 4x^-1

4/x^2 = 1

4 = x^2

x = +/- 2

c = +/- 2

By the mean value theorem, the function is continuous and differentiable, therefore some value c=2 where f ' (x) = the slope between (1,4).