Mathematician: "Let's use calculus to solve this." Engineer: "It's 16."

I think he uses Integral Calculus in grocery shopping.

Engineer: it's about 16. Add 25% contingency factor. Let's make it 20.

Next: solve a calculus problem using only arithmetic. 😁

You know someone is smart when they use Wolfram Alpha as a calculator

Binomial expansion of (x+y)^4 with x = 2 and y = -0.002 and picking the zero-th and first order: (x+y)^4 -> x^4 + 4x^3 y = 16 - 4*2*2*2*0.002 = 16 - 0.032 = 15,936 Which is exactly the same result as yours. In fact, that's exactly the same thing because you consider the derivative of (x+y)^4 with respect to y to be a constant, so you pick up to the first order. One could argue that you can use Taylor expansion of (2+x)^4 in x=0. You get (2+x)^4 = 16 + 32 x Plugin in x=-0.002 and you get 15,936 Oh, what a surprise, we also find the same result. How odd! In fact, all these methods are equivalent.

Approximately 16, you’re welcome . . . . . Don’t take it seriously lol

This is a more conplicated demonstration of tangent line approximations.

Excellent video. Also, it's really awesome to read all these comments offering other solutions as well! Math is so fun!

Really enjoyed this one. Thanks for refreshing my knowledge of differentials!

sir you are so great, the best thing I like about you is you always teach us happily which makes us understand maths easily, keep going sir never let us down, thankyou. H

This is very interesting! This approximation method is very precise and you can always experiment with other values as well.

First thing came into my mind was using Binomial Theorem; (2-dx)^4= 2^4 - 4. 2^3 .(dx) ........ so on. Rest of the terms include dx to the power greater than 1 so we can ignore them for any practical purposes since they will be negligibly small. So 16 -0,064 = 15,936 Thank you for this problem, was interesting to see. #YAY

The relative deviation of your approximation from the real value is 0.0006%. For practical purposes, this is usually neglectable.

I took a more direct approach using a bit of precalculus and knowing the binomial expansions of (a+b)^n: (1.998)^4 = (2-0.002)^4 a = 2 b = -0.002 = -2E-3 (scientific notation makes this process a bit easier) (a+b)^4 = 1a^4b^0 + 4a^3b^1 + 6a^2b^2 + 4a^1b^3 + 1a^0b^4 Powers of a from 0 to 4: 1, 2, 4, 8, 16 Powers of b from 0 to 4: 1, -2E-3, 4E-6, -8E-9, 16E-12 From there, plug in for a and b: 1x16x1 + 4x8x-2E-3 + 6x4x4E-6 + 4x2x-8E-9 + 1x1x16E-12 16 + 32x-2E-3 + 24x4E-6 + 8x-8E-9 + 16E-12 16 + -64E-3 + 96E-6 + -64E-9 + 16E-12 From there I just expanded the scientific notation into full decimal representations and added the positives together, then the negatives, and then I subtracted: 16.000 000 000 000 00.000 096 000 000 00.000 000 000 016 + --------------------------------------- 16.000 096 000 016 0.064 000 000 000 0.000 000 064 000 + ----------------------------------- 0.064 000 064 000 16.000 096 000 016 00.064 000 064 000 - ------------------------------------ 15.936 095 936 016 Exact value without having to manually multiply 1.998 by itself four times over and having to waste time with long-form multiplication. c: Although your method is a lot more eloquent, a whole lot faster, and if you're just doing quick back of the envelope math for a crude engineering calculation just to get a quick idea of what's going on or because your tooling just isn't that precise anyways... it's perfectly A-O-K to use.

Is it only me who heard *Doraemon* tune in the intro? Btw thanks for this amazing video

Excellent refresher, sir. Thank you!

This is really interesting how calculus can be used to solve such problems!

This is an excellent explanation for local linear approximation for anyone who has basic knowledge of derivatives

Awesome Video, Really Informative and Useful.Thank you so much.

(2-0.002)^4=(2-0.002)^2^2 ~(4-0.008)^2 (note: 0.002^2 is too small, we regard it as 0) ~(16-0.064) (note: 0.008^2 is too small, we regard it as 0) =15.936

This video explain so much to me that wasn't even to do with the video. Thankyou

Cal 2 pays off! No kidding. Great video, it's great to see application of calculus, love how alive you look and how the brain is being used!

Your videos are addictive. I enjoy these ingenious connections and tricks you come up with. Keep it up:) Much love from Kent State Uni

A lot of people are making too much of how this particular case lends itself to a variety of approaches. But the derivative approach can be used for just about any situation where the function is differentiable. For example, back in physics class, we used to calculate very small time dilation effects (which involved square roots of differences of squares) by differentiating the time dilation function and using that to calculate the delta. Also, this ties into the Taylor Series, which can be used to approximate complicated functions with polynomials: en.wikipedia.org/wiki/Taylor_series

I just wonder why didn't my teacher tell this very basic question when we were being taught calculus (it's been more than even a month since we started doing differentiation). Thanks to u I am able to understand this concept more!

I loved it, beautiful

I converted 1.998 to 999/500 then squared it in my head to 998,001 / 250,000 which is about 3.992. Notice how the delta of 0.002 was made into a new delta of 0.008 so there is a cube factor in there (2*2*2 = 8). So by squaring the intermediate result again, the new delta should be 0.064 (8*2*2*2).

Thank you for excellent explanation

Love it. Thank you

That's so cool!

Another excellent explanation....

Awesome. Now I understand the usage on differential calculus.

love you so much sir...you are my inspiration... love you sir... Good bless.

I figured out what you were doing halfway in and just kind of reveled in the genius

That what we call him The mean value theorem , thanks for teach us .❤️

Loved the video!

Amazingly clever.

I like it. It's a very good way to use calculus

Binomial Theorem can also solve it. (2-0.002)^4=16-4(8)(0.002)+6(4)(0.000004)-4(2)(0.000000008)+(0.000000000016) =16-0.064+0.000096-0.000000064+0.000000000016 =15.936095936016

Good teaching. Thank you, Sir.

Here in Poland we can't have calculators on exams at university so I've learned this on the beginning. And this is simple example. I remember people who was so angry with professor :) very useful approach not only for power. Try to calculate 4th root of 1558.57 without this method.

It can even be made simpler: To square x-a you get approximately x^2-2ax once that a is very small and so a^2 will be smaller. In our case, we get 4-2 . 2 . 0.002 which is 4 - 0.008. Repeating (because we want the fourth power), we get 16 - 2 . 4 . 0.008 = 16 - 0.064.

This is how I've always done mathematics since I began learning numbers. Never showing my work other than a couple of numbers I needed to remember along the way was always a problem in school.

Clever tool that I can use in computing harder examples. Those in the comments section missed out the point. He knows he can use algebra in breaking apart the given but what he wants to teach you is to approximate a function using differentiation.

Based upon your previous video, I thought you might do a video like this one! :)

This is beautiful.

Very applaudable, sir :) thanks!

Much better explanation than my prof gave me. Thanks!

brilliant video

Yo, I've learn this thing b4 but don't know a thing, thanks for clearing this up after 1 year :)

It's amazing!!!!... And have a request can you make a video on different graphs.

You get pretty much excited when doing math, and that's great!

It easier to use the linear approximation using first two terms in taylor series L=f(a) +d/dx f(a) (x-a)..... 1 where a is constant, and L stand for linear approximation of the original function f(2)=2^4=16......2 d/dx f(2) =4(2)^3=32.......3 Substitute 2 and 3 in 1 L=16+32(x-2) ....4 SUBSTITUTE x=1.998 in 4 implies L=15.936

Well explained 💪💪💪

I was thinking about this a little differently... In general, we have: (a + b)^n = C0*a^n + C1*a^(n-1)*b + ... + Cn*b^n, where C0, C1, ..., Cn are the binomial expansion coefficients. If 0 < |b|

Thank you so much

Really nice one man !! Love from India

Interesting application

Love the music in the end! #YAY

Beautiful

I love your teach ❤️️

I noticed that delta x / x is 1 per mille (1 part per thousand) I know that the proportion is multiplied by the power, so looking for delta y approx 4 per mille. 4 x 16 is 64 so have to subtract 64/1000 from 16. 15.936. Same as your answer but using an extra short cut

I read this in my high school calculus class

I like. Thank you. Reminds me of Taylor series + error estimation video I seen up here 😊

thank you awesome

Very cool thanks

This is equivalent to computing the linear approximation of x^4 around x=2, which is y = 32x - 48.

Great and useful video.

Thank you sir

If you wanna use approximation by calculus, it's better to use the form (1+x)^n ~ 1+ nx, where x

I learnt it today. Great. I learnt why I forgot calculus. Must be for good.

Hey sir Vishal from India thank you for this lesson it help me lot in my mathematic now I can easily solve question like these

My first thought was Taylor series, which would essentially give you the same result (first order approximation) . When you wrote down 2-0.002 I thought you might use the Binomial theorem. But your way of explaining it is nice for students that don't know about Taylor series yet.

I used Local Linearity principle and found the equation of tangent line at 2 using derivatives and then calculated function value of tangent line at 1.998 to have a better approximation of this problem

Holy fuck.... finally something does sense 4 me about why calculus is needed

It’s beautiful how the universe works.

Wow!!👍👍👌👌👌

FUNtastic, thanks

That reminds of high school, such a good nostalgia

I think using the binomial theorem and just doing the first two orders of it would give you the exact same answer you had, but with it you can got even more exact, if you want to do it for some reason

badass ... thanks for the video .... I love math 🙂✌️

Amazing

Wow you made the calculation more complicated kudos

Ths iz bheri helpful. thenkz.....heh heh

#yay best channel ever

Interesting. Thanks.

Thanks

You can use the formilation of taylor and you well get an exact value becouse x^4 is defferentiable 4 times

Awesome

I used this before watching the video in my head...

Taking 2 in factor, we have (1-1/1000)^4, approximately 1 - 4*1/1000, which gives immediately the correction to 16 : -64/1000. Your interesting demo gives the shortcut to the Taylor dev. on first order...

perhaps a shorter rouute is to transforn dy=4x^3 dx into dy/y = 4 dx/x dy= y( 4 * -0.001) dy =16(-0.004) Y = 16- 0.064 =15.936

The same procedure as considering a central value(2) and the error(0.002). The bottom limit gives your solution.

There is a nice relation to statistics. This way showed in the video is used in "propagation of uncertainty" in physics and statistics. It's called "variance".

Still an exposure problem - shifting light/dark depending on what the camera judges the white balance to be. arm to equation, screen lighter arm away, darker. lighter, darker, lighter, darker - sort of distracting

Working out this sum with Calculus is damn easy... - An IITJEE aspirant 😂

L(x) = f(2) + f’(2)(x-2) near x=2. I learned that a few weeks ago in my calculus class. L(1.998) = 16 + 32(-0.002) L(1.998) = 16 - 0.064 L(1.998) = 15.936

(2000-2)^4 use Pascal's triangle to easily do this, and then whatever you get move the decimal 12 places to the left

please do probability videos! i struggle with it :(