Important Theorems of Calculus

In a Nut Shell: Several theorems are basic to the understanding of elementary calculus.

They include Rolle's Theorem, the Intermediate Value Theorem, The Mean Value Theorem,
and the Fundamental Theorem of Calculus. This last theorem is especially important since it

establishes the connection between differential and integral calculus.

Rolle's Theorem:

Let f(x) be a function that satisfies the following conditions:

1	f is continuous on the closed interval [a,b]
2	f is differentiable on the open interval (a,b)
3	f(a) = f(b)

then there is a number c in (a,b) such that df(c)/dx = 0

Example:

Find all numbers c in the interval given that satisfy Rolle's theorem.

f(x) = x³ ˗ 3x² + 2 x + 2 with interval [0, 1]

Note: f(x) is both continuous and differentiable

Also f(0) = 2 and f(1) = 1 ˗ 3 + 2 + 2 = 2

df/dx = 3x² ˗ 6x + 2

f ' (c) = 3c² ˗ 6c + 2 = 0

c = [ 12 ± √ (36 ˗ 24) ] / 6 = 2 ± √3 / 3

The root c = 2 + √3 /3 is outside of domain.

The root c = 2 ˗ √3 /3 is within (0,1). (result)

Click here to continue with other important theorems.

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