Mean value Theorem and l'hopital's Rule

Calculus is one of the most important chapters in mathematics and it has divided into two types. There are Differential calculus and Integral calculus. The mean value theorem and l'hopital's rule are the important tolls in calculus.In this article we shall discuss about the  mean value theorem and l'hopital's rule.

the mean value theorem and l'hopital's rule-The mean value theorem:

In calculus coure, the mean value theorem states that , given an arc of a smooth  differentiable (continuous) curve, there is at least a point on that arc at which the derivative of the curve is equal  to the "average" derivative of the arc.

if a function f(x) is continuous to the closed interval [a,b] and differentiable to the open interval (a,b), and then there is exist point c in (a, b) such like that, the secant connecting the endpoints of the interval [a,b] is equal to the (parallel) to the tangent ‘c’.

The mean value theorem:

f’(c) = f(b) – f(a) / b - a

When f(a) = f(b) is generally known as Roll’s theorem. In that case , we have f’(c) = 0.

The mean value theorem is states also in trms of slope, so that the number

f(b) – f(a) / b - a

in this case the slope line is passing on (a(f(a)) and (bf(b)), so that only the mean value theorem concludes that, there is a exist point  is     c `in` (a,b)

So, the tangent line is parallel to the line passing through on (a(f(a)) and (bf(b)).

Example problem in the mean value theorem and l'hopital's rule:

Problem: find a value of c and satisfied mean value theorem for f(x) = -2 x^2 + 6x +1 on that interval (2, -2)

Solution:

F(x) is polynomial function, so let us evaluate f(x) at x= 2, and x= -2

F (2) = -2(2) ^2 + 6(2) +1 = 5

F (-2) = -2(-2)^2 + 6(-2) +1 = -19

Evaluate [f (b)-f (a)] / (b-a)

= [-19 -5] / (-2 -2)

= -24 / -4

= 6

Let us going to find f’(x)

f’(x) = -4x + 6

Now we have to go for equation f’(c) = [f (b)-f (a)] / (b-a)

-4c + 6 = 6

Solve for c and get value of

C = 0

I am planning to write more post on Alternate Interior Angle Theorem and Central Limit Theorem Example. Keep checking my blog.

l'hopital's rule:

It is a Procedure for differential calculus of evaluating and indeterminate forms such as 0/0 and 8/8.  When the result from attempting to find a limit. It states that the limits of f(x) / g(x)  is indeterminate, when it under certain conditions obtained by evaluating the limit of the quotient of derivatives of f and g.it is also called as Bernoulli's rule.

lim  f'(x) / g'(x)

has finite value( limit) is ± 8 , then

lim f(x) / g(x) = lim f'(x) / g'(x).

Example problem in l'hopital's rule:

Problem: Using l'hopital's rule , evaluate the  limx->`oo`   e^x / x

Solution:

`lim_(x->oo)` x / e^x

`lim_(x->oo)` 1 / e^x = `oo`

e^x / 1 =  `oo`