Linear Approximations and Differentials

Linear Approximation

In this video, I explain the concept of a linear approximation, which is just a way of approximating a function of several variables by its tangent planes, and I illustrate this by approximating complicated numbers f without using a calculator. Enjoy!

[NPRojas]

Great video!

Making maths easy and funny.

[dhunt]

Cool! Since I am on a deserted island (in my own home...), I do have a calculator handy - so I tried L(1.01, 4.01) and it has an error of less than .000014. Nice!

[txikitofandango]

Great video, I get a really great boost to my intuition about approximating functions of two variables here. I understand on some level the connection between the equation of a tangent line y - y1 = f'(x)(x-x1) and the tangent plane equation with the partial derivatives at 4:50 but I would've liked to see a little more explanation here. Anyways, I love your videos, always feel happy when I see a notification for a new one.

[GhostyOcean]

Can you do a video on quadratic approximations? My vector calc and calc 3 classes never did anything higher than linear approximations.

[davidseed]

when i did my engineering degree 1972, we did this all the time. for small percentages that are multiplied just add the percentages. powers become multipliers.
so (1.01)(4.02) becomes 4(1.01)(1.005) = 4 (1.015) taking sqrt gives 2 * (1.015)^0.5 = 2*1.0075= 2.015.

proof (1+a)(1+b) = 1+a+b+ab but ab is small about 1% of 1% and is ignored.
(1+a)^p = 1+ap + higher order terms which are ignored.

[salvatoregiordano]

Your videos are very helpful! Thank you so much

[eliyasne]

2:23
Actually, that doesn't look like the real thing.
The real thing is zero on (0, y) and (x, 0) and steadily increasing as you go deeper into the first quadrant.

[fmakofmako]

I tried doing the linear approximation about (1.01, 4.04) and got 2.15. I assumed it should have been closer than the approx for (1, 4), but they get the same approximation.

Edit: approximating about (1.005, 4.02) also has the same result. Interesting.

[tmstone]

2:21 That's not what f looks like. You rotated it around the (x, y) axis while it should have been rotated around the z axis. The idea of the explanation isn't impacted by this so I'll give you a pass😉

[williamadams]

At 10:37, does it mean that if the angle between plane L and the z-axis is “small”, then the approximation will be less accurate?

[jose-victorcabalu]

Hey wait, I know this guy. He's super cool

[tretyakov]

But we could solve it much easier. Sqrt

[kennethgee]

lol time index 3:02 modulo my bad drawings. Now that is an epic statement

[aarontierney]

Debated whether I should watch the vid... but I kind of wanted to see it.