Cauchy's Mean Value Theorem: Visual Proof

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Recently I was asked whether I could go over a visual proof of the Cauchy's Mean Value Theorem, as I had done for the Lagrange or simple version of the Mean Value Theorem (MFT). This was a very interesting question so I decided to go ahead and go over the graphical visualization of the theorem. In this video I show that the Cauchy or general mean value theorem can be graphically represented in the same way as for the simple MFT. The only difference is that the horizontal axis is not x, but a more general function of x, g(x). This difference causes the formulation to be more general, and the instantaneous slope to be formulated in the more general definition of the derivative, as shown in my last video. This is a very interesting topic to understand, but make sure to watch my earlier video on the general definition of derivative to get a better understanding of this!
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Опубликовано:

17 сен 2024

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Комментарии : 67

@mes 6 лет назад

I don't always go over Cauchy's Mean Value Theorem but when I do I usually visually prove it using a general function as the horizontal axis ;) View Video Notes on Steemit: steemit.com/mathematics/@mes/cauchy-s-mean-value-theorem-visual-proof

@azulmarina92 6 лет назад

This explanation was really useful, I was searching a geometric interpretation of Cauchy MVT and it was so hard to find something besides Lagrange MVT. Also, the questions and answers in the comments were as useful as the video. Thank you!

@ap9386 7 лет назад

I don't want to sound rude, but it's "koh-SHEE", not "KAU-chee", it goes after the French surname of the mathematician that formulated the theorem. Excellent explanation by the way!

@mes 7 лет назад

Thanks! And I'll try to pronounce it correctly next time haha :)

@321123moni 7 лет назад

Excellent video. It helped me lot with my exam. Thank you :)

@mes 7 лет назад

Sweeeet! Glad my video helped you on your exam! ʕ •ᴥ•ʔ

@pruthveshmandle4978 Год назад

Great video. If you would've expressed the two functions in parametric form, then the explanation could've seemed much easier. I had trouble finding what f'(c)/g'(c) meant, turns out it was just dy/dx (parametric differentiation)

@luthfibalaka7558 3 года назад

What a really good explanation, It really helped me understand where the equation come from. Thank you, sir 👍

@ejsafara456 Месяц назад

oh my, thank you very much for expaining and drawing it! i had something in my notes, but couldnt figure out what it actually meant XD

@satwantrajpurohit4145 7 лет назад

tysm sir . ✌ now it is easier for me to remember these.

@mes 7 лет назад

You are welcome!! ʕ •ᴥ•ʔ

@sangarg4611 4 года назад

The graph xou drew in that at y axis there should be f(g(a)) and f(g(b)) I think so because we input the values in x axis at function at y axis.

@sameersahil2757 2 года назад

No dear! Your Domain set is on z-axis you choose values from this axis and they fall on x-axis and y-axis.

@HarshRajAlwaysfree 3 года назад

thanks that was an interesting way to look at the equation

@kkmathspoint1231 6 лет назад

But it will 2 sperate function f and g How can we say x axis point as g(x)

@vaibhavsrivastava7732 5 лет назад

Thats what i thought

@diegowang9597 5 лет назад

Just wonder how is Mean Value Theorem shorthanded as MFT, but not MVT

@mes 5 лет назад

ahhhh my mistake! I did my Masters on Mature Fine Tailings (MFT) while in University, so it is an accidental habit to label things similar! :)

@jamesrobertson9149 4 года назад

this is very helpful. thanks.

@shuddhoshawttoroy6257 3 года назад

Extremely good explaination.

@amiyancandol4499 4 года назад

Can someone tell me what is the function g(x) is it like a double derivative or like any other function

@sripavanch4085 7 лет назад

how can one plot f(X) on y axis and g(x)on x axis ? It is not dependent on g(X)...f(x) only depend on x.

@mes 7 лет назад

Thanks for asking. You can choose the axis to be anything you want. Even when the x-axis is just x it can be viewed as the function g(x) = x. Regardless, both f(x) and g(x) are dependent on x, so if none of them are equal to exactly x, you would need a separate table or 3D axis showing the x values.

@mes 3 года назад

@@fajarbarari779 Thanks for asking. I think that the triple bar ≡ in that proof just means "if and only if". That is, "The classical MVT is a special case of CMVT if and only if g(x) = x." You can read more about how the triple bar is used here: en.wikipedia.org/wiki/Triple_bar

@liwen9244 7 месяцев назад

@@mesSo is g(x)=x also one of the condition for MVF to be true?

@mes 7 месяцев назад

@@liwen9244 It is a condition only for the basic MVT.

@liwen9244 7 месяцев назад

@@mes So what if for general MVF? Like when you plugs x into g(x), is the y-coordinate that you get from that point on the curve still equal to f(x)? Thanks for replying so fast!!

@surajnongmaithem2019 4 года назад

Sir one more for Lagrange's theorem. Please

@kevind3946 7 лет назад

I'm not even taking calculus yet, but you make it sound so easy! thanks!

@mes 7 лет назад

Sweet glad I can help! Calculus is similar to any other language, so it's easy once we think of it as a language. If you ever have the time, I suggest listening to this amazing math lecture: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-NnVubBrATIU.html

@MikeJenson Год назад

Pls help. I'm confused. Your diagram looked like a "C" on its side. There are points at which the slope is vertical. Does this not mean that g(x) is not changing, hence g'(x) is zero. Does this not contradict the hypothesis.

@mes Год назад

Yes there are points where the slope is vertical. But this doesn't mean that g(x) is not changing. It just means that g(b) - g(a) is approaching 0. The derivative g'(x) is unrelated to it.

@ghegogago8297 7 месяцев назад

Sorry, but if the graph f(x) over g(x) isn't a function (note the vertical slope) how can we apply MVT? (Because it's what we are implicitly doing to assert that there must be a point c where the slope is blah blah, right?)

@mes 7 месяцев назад

The condition for the MVT is that g'(x) is not zero, hence it only applies when the slope is not vertical.

@ghegogago8297 7 месяцев назад

@@mes Do you mean the Lagrange MVT?

@mes 7 месяцев назад

@@ghegogago8297 Both. The Lagrange (basic) MVT is just g(x) = x, and g'(x) = 1 aka not zero.

@rs-sl1tn 7 лет назад

Could u please verify Cauchy's theorem graphically for ℯ^x & ℯ^(-x) together in the interval (a, b)

@azulmarina92 6 лет назад

download gnuplot

@klapel3794 2 года назад

The way he pronounced Cauchy almost killed me

@rajibbehera-fg1gj Год назад

Grateful❤ to you sir

@mathclub3669 2 года назад

But we discuss two functions and you draw graph for single function only.

@mes 2 года назад

The graph is of 2 functions.

@a1mratz 3 года назад

This helped a lot! Is there a link for the "previous video" you are mentioning in the video? Thanks a lot!

@mes 3 года назад

Thanks for asking. The link is in the video description in the "related videos" section.

@koushikshomchoudhury9108 7 лет назад

for function f here, c is greater than b! -_- how can this be true?

@mes 7 лет назад

Hi Koushik, thanks for asking. f(c) can be greater or even less than f(b). That is not a requirement of the Cauchy's MFT. The only requirement is that a < c < b. Also, from the visual proof I believe g(a) < g(c) < g(b) as well, but I will need to double check this to confirm.

@koushikshomchoudhury9108 7 лет назад

i am sorry! i mistook f(c) for c. Apologies.

@diegowang9597 5 лет назад

What if c is not between a and b? It's possible on the f(x) against g(x) graph, because the slope of the curve could be greater than the average slope between a and b, and start to decrease for g(x) > g(b)

@mes 5 лет назад

It's not a matter if it's possible, but if it always applicable. If c is not between a and b, then there are scenarios where the derivative is NOT the average slope; for example if c is on a curve point with negative slope but the curve from a to be is straight positive slope.

@minwooyoo1167 2 года назад

In the graph, I can't find why c have to be located between a and b. I think it is the most important part of the theorem as it is in normal MVT.

@minwooyoo1167 2 года назад

Oh I just figured it out. It seems that the curve is like a journey of (g(x), f(x)) as x moves from a to b. So, if we find the same slope on the curve, no matter where it is, it could be where c between a and b is. Thanks for the video.

@YeshaniNethmi-q6o 2 месяца назад

@andreutormos7210 3 года назад

How come f(x) have two possible values for the same input, say g(a) ???

@mes 3 года назад

f is a function of x and not g. There is only one value of f(x) for each x value. At g(a), the value of x = a and another variable, let's call it z. For example, a sine function has g(x) = sin(x) = 0 at x = 0 or pi. Thus the two values of f at g(a) are f(x) and f(z). Hope this helps!