Lecture 11: Lagrange Mean value theorem. Geometrical and Physical interpretation with an example

One of the very important theorems in calculus and its Lagrange's Mean Value theorem.
We will try to understand Lagrange's Mean Value Theorem. We will also see its Geometrical and Physical interpretation. At the end we will see a nice example related to Lagrange's mean value theorem. The example will act as an application to Lagrange's Mean Value theorem.

[@rohanzende.coepian2506]

Nice explanation, sir 🙏

[@DrMathaholic]

Thank you Rohan 😊

[@Bikrampaswan01]

No words 🙏🙏🙏

[@DrMathaholic]

🙏😊

[@KhiladiGamer93]

sir example at last was really cool......

[@DrMathaholic]

Thanks 😊

[@AJ-yu2ur]

Satisfy the hypothesis of the Mean Value Theorem on the given interval and which does not? Give reasons F(x) ={x^2 - x [-2 -1] {2x^2 -3x -3 (-1 0]

[@DrMathaholic]

Both are polynomials and they are continuous and differentiable. The function only breaks at -1. So we need to check at at -1. Lim x -> -1 from left side gives 2 whereas lim x->-1 from right side gives 2. So f is continuous over the complete interval. Now take respective derivative, we get 2x-1 and 4x-3. Again at -1 , they left and right are not equal so at -1 it's not differentiable

[@abhishekkajirnekar14]

Sir, how to check given function satisfies hypothesis of mean value theorem. Ex. f(x) = x^2/3 [-1,8]

[@thorrr7613]

Same question , how to solve for this example F(x) ={sinx/x [-π 0) ={ 0 x=0

[@DrMathaholic]

It's a polynomial so it's continuous and differentiable function. So it satisfies LMVT conditions

[@abhishekkajirnekar14]

@@DrMathaholic but in answer given that it doesn't satisfy hypothesis ( Thomas calculus 14 th edition)

[@DrMathaholic]

@@thorrr7613 sinx / x is continuous everywhere except at 0. For non zero points, numerator and denominator both are continuous functions so their ratio is continuous BUT x=0 , f(0) is defined to be 0 where as lim x->0 f(x)=1. Thus f(0) Is not equal to lim x->0 f(x) i.e actual value of f at 0 is not same as limiting value of f as x approaches 0. Thus f is not continuous at 0

[@DrMathaholic]

@@abhishekkajirnekar14 it's x^{2/3} or (x^2)/3??

[@specialtech7386]

Wonderful ✨😍

[@DrMathaholic]

Thank you 😊

[@pradipbhandri]

Thank You❤

[@DrMathaholic]

Welcome 😊

[@ananyagupta4437]

Sir i can't understand the instantaneous change which is f´(c) in this video.... It will be very helpful if you will make it clear 🙏

[@DrMathaholic]

See if this helps:: kzbin.info/www/bejne/hXuuf3d4qrqkmpo

[@ananyagupta4437]

Thank you sir, it was helpful

[@thorrr7613]

Satisfy the hypothesis of the Mean Value Theorem on the given interval and which does not? Give reasons , how to solve for this example ? F(x) ={sinx/x [-π 0) ={ 0 x=0

[@DrMathaholic]

@THORRR sinx / x is continuous everywhere except at 0. For non zero points, numerator and denominator both are continuous functions so their ratio is continuous BUT x=0 , f(0) is defined to be 0 where as lim x->0 f(x)=1. Thus f(0) Is not equal to lim x->0 f(x) i.e actual value of f at 0 is not same as limiting value of f as x approaches 0. Thus f is not continuous at 0

[@thorrr7613]

Thank you sir 😊

[@DrMathaholic]

@@thorrr7613 wlc...

[@engineeringstudent1959]

Sir there is one way of not getting caught by the Police we can initially drive at faster speed and later at a slower speed and when avg will be calculated then it will be less than 70kmph. In this way one can enjoy fast driving also and won't be caught as well😂😂😂

[@DrMathaholic]

Hahahaha..nice one . u enjoy fast driving then take hault or something..try to reach next toll, say, in 2 hrs n 20 mins..😉 then ur avg speed will b less... 😂

[@engineeringstudent1959]

@@DrMathaholic yes sir😂😂

[@AkashPatil-wo8jh]

i think ...sir apka chalan kata he .....tabhi me sochu itna hatke example kaise

[@DrMathaholic]

Hahaha.. Yes , 1000 rs 😔😔 that's y one should drive slow 😂

[@viraj6099]

Sir mast example tha 😂.

[@DrMathaholic]

😂😂😂 enjoy 😀