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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28 thoughts on “What is linear approximation?

  1. 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.

  2. 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!!

  3. 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 ??

  4. 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.

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

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

  7. 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…

  8. 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…

  9. 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! 🙂

  10. 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?

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