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Applying Rolle's Theorem with Trigonometry [Ex. 1]

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STEP by STEP guide to The Mean Value Theorem

No jokes about the theorem being mean pls.

This is another common theorem presented in first-year calculus courses. The theorem might sound scary, but it’s actually quite simple.

We observe a function f that is continous on [a, b] and differentiable on (a, b). If I draw a line from one end point to another, f(a) to f(b), find the slope of that line, and call it m, the theorem states that somewhere on the interval (a, b), the function f will have the same slope as m.

The proof is also quite elementary and simple. In order to prove this theorem, we need to explore Rolle’s theorem. I will prove Rolle’s theorem in another post. You can find the theorem stated in the proof.

STEP by STEP guide

Draw a line connecting the end points of the interval.

Find the equation of the line using y=mx+b

Create a new function g(x) = f(x) - the equation of the secant line y (Assuming f(x) > y). We are looking at the space between our function f and the secant line.

We notice that at the end points, g(x) = 0. This is because f(x) = y at the end points, so the space in between them = 0. Therefore Rolle’s theorem applies.

so there exists a c in (a, b) where g’( c ) = 0.

WE TAKE THE DERIVATIVE OF g(x) and set it to be equal to 0!!!!

REARRANGE THE DERIVATIVE A LITTLE BIT AND WE GET THE THEOREM!!

congratulate yourself by taking a nap.

bam. Now you know.

#math #mathematics #rolle's theorem #calculus

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A Monotonic is a function between ordered sets that preserves or reverses the given order....so lets watch and understand the concepts of Monotonic Function.

"In this video you will came to know about increasing and decreasing function & Rolle's Theorem ,Lagrange's Theorem with Extraclass.com"

#Rolle's Theorem #Lagrange's Theorem #physics #decrease #increase #monotonic function