# What is linear approximation?

► My Applications of Derivatives course:

0:00 // What is linear approximation?

0:44 // When do you use linear approximation?

1:28 // Estimating square roots using linear approximation

5:23 // Estimating trig functions using linear approximation

6:37 // How to find the error in a linear approximation

7:48 // Summary

Linear approximation, or linearization, is a method we can use to approximate the value of a function at a particular point. The reason liner approximation is useful is because it can be difficult to find the value of a function at a particular point.

Square roots are a great example of this. We know the value of sqrt(9); it’s 3. That’s easy to figure out. But we don’t know the value of sqrt(9.2). We can guess that it’s a little bit more than 3, since we know that sqrt(9) is 3, and 9.2 is a little bit more than 9, but other than that, we don’t know how to find a better estimate of sqrt(9.2). That’s where linear approximation comes in to help us.

Since we’re dealing with square roots, if we imagine the graph of the function sqrt(x), we know one point on that function is (9,3). If we find the tangent line to the function sqrt(x) through the point (9,3), then we can see that, since the tangent line is really close to the graph of the function around the area of (9,3), that the value of the function and the value of the tangent line will be pretty close to each other at x=9.2.

So to get an estimate for sqrt(9.2), we’ll use linear approximation to find the equation of the tangent line through (9,3), and then plug x=9.2 into the equation of the tangent line, and the result will be the value of the tangent line at x=9.2, and very close to the value of the function at x=9.2.

That’s why linear approximation is so helpful to us, because it’s a quick, simple method that let’s us estimate a value that would otherwise be very difficult to find.

Music by Joakim Karud:

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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. 😉

Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”

So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student—from basic middle school classes to advanced college calculus—figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here:

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hi Krista! can you help me with this? Q1. Write down the linear (tangent line) approximation to y = e x/ 2 at the

point x = 0, and sketch the graph of this function and the tangent line. my email olhy1019@gmail.com thanks in advance.

Sal Khan (who is a genius) had me going in circles with this. Thanks, Krista, for the clarity.

I love you Krista😭. Thank you so much

This video was very helpfull to me

Thank u so much for this lesson

this video is AMAZING thank you soooo much for the brilliant video, I really appreciate the effort.

fantabulous, excellent, simply super, wow… love u r teaching…

Super 😍

I usually don't comment on videos, but for this one, I had to! Brilliant Job!

The sheet I got from my prof explaining this just looked like a giant mess of squiggly lines to me. This broke it down into individually presented little pieces, and was super helpful. Thank you thank you thank you!!

so basically if you dont have a calculator then its useful?

do you like doge

Love the video but I am missing the jump from understanding why pi/3 is close to 1.05 ??? ( 5:41 ) how do I know that ??

Well explained vids with good quality

Keep it up !!

KK, you explained this better in 9 minutes than a 200 dollar textbook and a college professor paid 200k/year at my local college. You are living proof that math education needs to modernize itself. James Stewart's textbook is damn garbage.

Amazing content! Very simple yet effective explanations. I wish your channel had more subscribers though! You're so underrated!

Where was this video in 2017 when i needed it desperately.At last i Understand approximation.

wow, this is astonishingly high quality for a simple calculus video, great work

this was sooo helpful! you are a gem

Good! But too fast for a slow learner like me

This is wonderful

First video I've ever seen of yours, nice job. But you look like a prison inmate in the thumbnail haha

I actually do love all you videos for they are simple and basic and presented in a smooth step by step fashion… as in this one if I cannot follow them and find them logically false i cannot rely on them… and that makes me sad bec i love you ma'am…

hello dear ms. krista king, in this video what you call an error and what you calculate for error is not the exact difference between the real value of f(1.05) and the approximated number found for f(1.05)… are we clear about this ma'am…

production quality is outta this world!

Thank you so much!

Delta y doesn't give the error in the estimation, it gives the difference between the y values of two points, one being the point where you took the derivative, and the other being the point on the tangent line that has the same x value as the point you are trying to approximate. This is easier to see on a graph, but I only have text available. 🙁 I think where you might have gotten things mixed up is when you used f(x_2) to denote the value of the function at x_2 (see 6:54), but the equation you use it in implies that f(x_2) lies on the tangent line of the function at (x_1,f(x_1)). Therefore, f(x_2) is really the approximation of the function at x_2 and so f(x_2) – f(x_1) = Delta y is really just the difference between your estimation and the point where you took the derivative.

Thanks for working to make high quality, easy to understand math videos! 🙂

Hi Krista, just saw your tutorial. It's amazing. The creative way by which you were passing over this information is fantastic. But I have certain problems with your notations, don't you think linear approximation is more precise word you should use instead of linear interpolation as by interpolation we mean to find approximate function in a given range when we have information about function in that range. You aren't really using any range here, you are using information at a point to find information on another point.

Second point is when you calculated error, how can that be an error?