Mean value theorem function, Mathematics

Assignment Help:

Mean Value Theorem : Suppose f (x) is a function which satisfies both of the following.

1. f ( x )is continuous on the closed interval [a,b].

2. f ( x ) is differentiable on the open interval (a,b).

Then there is a number c such that a < c < b and

f ′ (c ) = f (b ) - f ( a ) /b - a

                    Or,

f (b ) - f (a ) = f ′ (c ) (b - a )

Note as well that the Mean Value Theorem doesn't tell us what c is. Only it tells us that at least there is one number c that will satisfy the conclusion of the theorem.

Also note that if it weren't for the fact that we required Rolle's Theorem to prove it we could think of Rolle's Theorem as a special case of the Mean Value Theorem.  To illustrates that just suppose that f ( a ) = f (b ) and then the result of the Mean Value Theorem provides the result of Rolle's Theorem.

Before we see couple of examples let's think about a geometric interpretation of the Mean Value Theorem.  First define

 A = (a, f ( a )) and B = (b, f (b )) and then we know from the Mean Value theorem that there is a c such that a < c < b and that

 f ′ (c ) = f (b ) - f ( a ) /b - a

 Now, if we draw in the secant line connecting A & B then we can know that the slope of the secant line is,

                         f (b ) - f ( a ) /b - a

Similarly, if we draw in the tangent line to f ( x ) at x = c we know that its slope is f ′ (c ) .

What the Mean Value Theorem described us is that these two slopes have to be equal or in other words the secant line connecting A & B and the tangent line at x = c has to be parallel. We can illustrate this in the following sketch.

780_tanglent line.png


Related Discussions:- Mean value theorem function

Pairs of straight lines, The equation ax2 + 2hxy + by2 =0 represents a pair...

The equation ax2 + 2hxy + by2 =0 represents a pair of straight lines passing through the origin and its angle is tan q = ±2root under h2-ab/(a+b) and even the eqn ax2+2hxy+by2+2gx+

Algebra 2 Appendix F, I have an algebra assignment I need help with, you ha...

I have an algebra assignment I need help with, you have helped me before.. I need the work shown.

Trigonometry, trigonometric ratios of sum and difference of two angles

trigonometric ratios of sum and difference of two angles

What distances from the two gates should the pole, A pole has to be erected...

A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that the differences of its distances from two diametrically opposite fixed gat

Find out the roots of the subsequent pure quadratic equation, Find out the ...

Find out the roots of the subsequent pure quadratic equation: Find out the roots of the subsequent pure quadratic equation. 4x 2 - 100 = 0 Solution: Using Equation

Business mathematics, explain how business mathematics in an inbu;it compo...

explain how business mathematics in an inbu;it component of a payroll package

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd