Linear Approximation and Differentials
October 7, 2026

[00:00:05] Somebody tell me where are these two things?
[00:00:09] Point-slope form. Okay, point-slope form. Yes, close. They are definitely point-slope form.
[00:00:26] But usually we don't put dy dx here. What do we normally put? M. M is the slope. Since I have dy dx or F prime, that's the slope of the tangent line.
[00:00:47] Okay. These are simply equations of the tangent line. Equations of the tangent line to y at x1, y1.
[00:01:20] So some function y. And why do I notice the function y for the top one? Because of this, is the derivative of this some function y, which is not the same function y.
[00:01:35] This notation is a little bit better on bottom because this is definitely the equation of the tangent line.
[00:01:54] To the function F at a comma F of a.
[00:02:09] It's just using two different notations. Both slopes are derivatives. So that's the slope of the tangent line. So these are the equations of the tangent lines.
[00:02:25] At either x1, y1 of this function y or a comma F of a. Same thing.
[00:02:33] What the function is this?
[00:02:36] Polynomial.
[00:02:38] Polynomial.
[00:02:39] Okay. Find the equation.
[00:02:48] The tangent line.
[00:02:49] Sub urban.
[00:02:53] One.
[00:02:54] X equals zero.
[00:03:01] Oh.
[00:03:02] It's all right.
[00:03:16] So what do we got to do first?
[00:03:22] So to find the equation of the tangent line, are we going to use this one or this one?
[00:03:37] The bottom, why the bottom?
[00:03:44] More importantly, this is not y equals, this is f of x equals.
[00:03:51] Okay, so we need to find f prime of a, the slope of the tangent line.
[00:03:59] To find f prime of a, we need to find what?
[00:04:05] f prime of x for any x value.
[00:04:08] Six x to the fifth minus thirty-five x to the fourth plus sixteen x cubed minus three x
[00:04:18] squared, plus six x plus two.
[00:04:24] There we go.
[00:04:27] Very simple, using a bunch of power rules and sum-and-difference rules.
[00:04:34] Okay, and then we need f prime of a, right?
[00:04:40] f prime of 0, 0 minus 0 plus 0 minus 0 plus 0 plus 2.
[00:04:50] f prime of 0 is 2.
[00:04:56] And the only other thing we need is f of 0.
[00:05:02] The y value.
[00:05:06] 0, 0, 0, 0, 0 minus 8.
[00:05:15] So y minus f of 0 equals f prime of 0 times x minus 0.
[00:05:29] What was the slope? Two.
[00:05:49] It's kind of ugly. It looks like it only has two zeros, two real zeros.
[00:05:56] How many complex zeros does this have?
[00:06:04] What's the degree of this polynomial we wrote down?
[00:06:07] Six.
[00:06:08] Six.
[00:06:11] Guaranteed to have six.
[00:06:14] There's a maximum of five turning points or minimums and maximums.
[00:06:23] Anyways, we found the slope of the, or sorry, we found the equation of the tangent line, which was what?
[00:06:30] Y equals, say again?
[00:06:35] Two x minus six?
[00:06:38] Minus eight.
[00:06:42] Okay, we see it's tangent right there.
[00:06:47] If I zoom in,
[00:06:53] you can't even tell the difference between those right there, right?
[00:06:56] They only touch at one single part. Right there at zero negative 8.
[00:07:08] Nearby x equals zero, when I zoom in, we're only looking at very small x values.
[00:07:15] Nearby, the two functions look exactly the same.
[00:07:19] They're just slightly off.
[00:07:20] We agreed? Okay.
[00:07:23] So this is a crazy function.
[00:07:42] Near x equals zero.
[00:07:48] This crazy function is approximately this function.
[00:07:56] f of x is approximately 2x minus 8, which is a very simple linear function.
[00:08:09] Okay.
[00:08:15] So,
[00:08:17] if we want to approximate
[00:08:24] f of, let's say,
[00:08:27] point one.
[00:08:31] Actually, let's say we want to find f of point one.
[00:08:37] It would take a lot of work, right?
[00:08:40] Point one to the sixth power minus seven times point one to the fifth power plus four times point one to the fourth power minus point one cubed
[00:08:52] plus three times point one squared plus two times point one minus eight.
[00:09:00] We're going to take a lot of work and probably need a calculator.
[00:09:05] Yeah. Okay.
[00:09:08] We'll plug that in in a second, but the approximation is very simple.
[00:09:15] It's we plug it into this function.
[00:09:18] Two times point one minus eight,
[00:09:22] which is what point two minus eight, which is seven point.
[00:09:31] Negative seven point eight.
[00:09:33] Seven point eight. Is that right? Yeah. Okay.
[00:09:38] So if we want to find values of f
[00:09:44] nearby the x equals zero, like point one is pretty close, point zero.
[00:09:50] It's very simple. We find the equation of the tangent line.
[00:09:57] It's simply plug that value into the equation of the tangent line.
[00:10:03] So this is called the is called linear approximation.
[00:10:09] Actually, it's called three different things.
[00:10:13] Linear approximation.
[00:10:19] Okay. We approximated some very complicated function with a simple linear function.
[00:10:25] That's why it's called linear approximation.
[00:10:27] Or it's called linearization.
[00:10:38] Why is it called that?
[00:10:42] We turn a nonlinear function into a linear function.
[00:10:48] So it's the linearization of the function. We turn it to a linear function.
[00:10:54] Okay. We linearize it. It's also called the tangent line approximation.
[00:11:06] Why is it called that?
[00:11:09] It's the tangent line.
[00:11:11] We're approximated with the tangent slope, or sorry, the equation of the tangent line.
[00:11:16] It all means the same thing.
[00:11:25] So this is it. Any function.
[00:11:36] Well, actually before I write that down, if we take this thing right here
[00:11:46] and solve for this guy, which is our linear equation,
[00:11:51] we get f prime of a times x minus a plus f of a.
[00:11:59] This is the equation of the tangent line.
[00:12:03] And this linear function here, y, is our linear function.
[00:12:11] Okay. So it is f of x is approximately f prime of a times x minus a plus f of a.
[00:12:30] This is known as the linearization of a function or linear approximation.
[00:12:34] Let's just call it linear approximation of f.
[00:12:45] But it's only the linear approximation of f at x equals whatever number a is.
[00:12:53] Let's find the linear approximation for the same function at x equals 1.
[00:13:05] Okay. At x equals 1, f of x is approximately f prime of 1 times x minus 1 plus f of 1.
[00:13:20] And we just have to find two things, the derivative at 1 and the function value at 1.
[00:13:28] It's only different than the previous problem.
[00:13:35] So where we're at? Let's do that up here real quick.
[00:13:44] This is going to take a little bit more work.
[00:13:50] f prime of 1 is 6 minus 35 plus 16 minus 3 plus 6 plus 2.
[00:14:00] Could somebody do that work?
[00:14:02] So f of 1, what did we say it was?
[00:14:07] Negative 6.
[00:14:08] f prime of 1 is?
[00:14:12] Negative 8.
[00:14:12] Negative 8.
[00:14:13] So our function is approximately negative 8 times x minus 1 plus minus 8 minus 6.
[00:14:30] You just want to keep it real simple.
[00:14:33] Negative eight x plus two.
[00:14:56] And let's approximate at.9.
[00:14:59] Is that close to 1?
[00:15:06] Oh.
[00:15:08] What is that negative.72 plus 2, which is 1 point?
[00:15:26] 1.28, is that right?
[00:15:39] Can somebody confirm my basics arithmetic?
[00:15:46] Is that negative 7.2?
[00:15:49] Yes.
[00:15:51] Is that negative 5.2?
[00:16:02] There we go.
[00:16:04] Sorry, it was y minus, or sorry, y equals, what did we get?
[00:16:11] Negative 8.
[00:16:22] Okay.
[00:16:33] So this is the approximation right here.
[00:16:36] Let me turn off the old one.
[00:16:39] Okay.
[00:16:39] Again, it's only the approximation near x equals 1.
[00:16:44] Okay.
[00:16:45] Every different x value, we have a different linear equation, different change in line, different linear equations.
[00:16:54] Oops.
[00:17:00] Okay.
[00:17:00] Let's do some more examples.
[00:17:01] Let's keep it simple.
[00:17:04] Again, our function.
[00:17:10] Just the point slope formula, solve for the y, it's f prime of a times x minus a plus f of a.
[00:17:22] This should say approximation.
[00:17:34] What is f prime?
[00:18:04] What is f prime?
[00:18:12] f of x is approximately one times x minus zero plus zero.
[00:18:29] Sorry, let's just say f of x is approximately x.
[00:18:38] This is our linear function.
[00:18:42] So sine of x is approximately just our function x near x equals 0.
[00:18:54] In fact, let's look at that real quick.
[00:18:56] This is kind of handy.
[00:18:59] For example, we're going to find sine of 0.2.
[00:19:05] We definitely need a calculator for that, right?
[00:19:08] Can somebody take out the calculator and tell me what that's equal to?
[00:19:13] It's approximately 0.2.
[00:19:16] Oh, got to make sure we're in what mode?
[00:19:19] It has to be in radians. Let's see what this is.
[00:19:22] Oh, it is.
[00:19:26] So sine of 0.2 is about 0.198.
[00:19:32] Slightly less.
[00:19:37] But 0.2 is pretty good approximation.
[00:19:40] It's kind of cool.
[00:19:44] It's a complicated function, right?
[00:19:45] Well, it's not that complicated.
[00:19:51] What's that like that?
[00:19:54] See if you can do this.
[00:19:55] Find the equation.
[00:20:00] Find the linear approximation at x equals 1.
[00:20:27] Okay.
[00:20:51] What's the derivative of the natural?
[00:20:58] Okay.
[00:20:58] Which is just one. That's right.
[00:21:06] And so that's the derivative.
[00:21:10] So f prime of one is simply one over one.
[00:21:15] And what is the natural log of one, everybody?
[00:21:18] Zero.
[00:21:23] So our function, ln x, is approximately one times
[00:21:33] X minus one plus zero.
[00:21:41] There is x minus one.
[00:21:43] Which is that line right there.
[00:21:54] So let's approximate ln. What's the number near one?
[00:22:01] One point two.
[00:22:08] That is 1.2 minus one.
[00:22:13] Let's see what ln of 1.2 is.
[00:22:28] Sorry, ln of one point two.
[00:22:36] So we got point two. It's really point one eight.
[00:22:40] It's a pretty good approximation.
[00:22:42] The farther we get away from our a value, in this case it was one.
[00:22:46] So the worse approximation.
[00:22:49] The closer we get, the better approximation.
[00:22:52] There is a way to calculate the error.
[00:22:54] We don't need to talk about that, but you don't need that for this.
[00:23:01] Actually going back here.
[00:23:08] Without actually plugging in the calculator, how can we tell if this is going to be an overestimate or an underestimate of the actual value?
[00:23:20] We should discuss that.
[00:23:22] Looking at the graph, how can we tell if it's going to be an overestimate or an underestimate?
[00:23:29] What's that?
[00:23:32] The tangent line is higher, greater than the function.
[00:23:38] So it's definitely going to be an overestimate.
[00:23:40] Let's say this is one point two.
[00:23:44] And we're going to learn later, because this is curving down, called concave down.
[00:23:54] We're going to be able to tell whether it's an overestimate or an underestimate.
[00:23:58] Next question, approximate.
[00:24:06] The square root of nine point three.
[00:24:15] Here, I just give you a value.
[00:24:21] So we're going to have to come up with two things.
[00:24:23] We're going to have to come up with a function and an A value where we can easily find the function value at that A value.
[00:24:37] So what function are we going to come up with?
[00:24:43] Let's go with the square root of X.
[00:24:47] And what A value?
[00:24:49] What value do we know the square root of nearby?
[00:24:52] Nine point three?
[00:24:54] Nine.
[00:24:56] So F prime of X is, what is that?
[00:25:03] One half X to negative one half.
[00:25:08] One over two square root x.
[00:25:15] F prime of nine.
[00:25:21] Is that one sixth?
[00:25:28] And F of nine.
[00:25:33] Square root of nine is three.
[00:25:39] So our function, square root x, is approximately
[00:25:43] f prime of nine times x minus nine plus f of nine.
[00:26:05] There's our linear approximation.
[00:26:09] Our linearization.
[00:26:12] Tangent line approximation.
[00:26:14] Square root of X.
[00:26:18] And now we plug in nine point three.
[00:26:22] Is that what I said nine point three?
[00:26:24] Yes.
[00:26:25] Okay.
[00:26:46] Point three.
[00:26:50] What is that?
[00:26:54] Tangent line.
[00:26:59] Three over sixty.
[00:27:07] One twentieth. Is that right?
[00:27:14] Change that to decimals. That's three point oh five.
[00:27:23] So that's a pretty good approximate.
[00:27:25] Well, I don't know how good it is, but we'll do that with some very simple math.
[00:27:33] Obviously all we got to know what derivatives are. All our rules.
[00:27:40] At least the power rule. Let's see, what is the square root of 9.3?
[00:27:51] What do we get? Three point oh five.
[00:27:55] Three point oh four nine.
[00:27:57] Pretty good.
[00:28:05] So linearization has everything to do with finding the equation of the tangent line to some function.
[00:28:11] Finding the derivative and plugging in the values.
[00:28:14] Let me draw you something real quick and then I'll explain it.
[00:28:28] This is some function.
[00:28:34] And it's some X value.
[00:28:39] This would be the tangent line.
[00:28:51] Okay.
[00:28:54] And if you have some other X value.
[00:29:21] Let me do different colors here.
[00:29:24] This right here is what's known as delta X.
[00:29:28] The change in X from A.
[00:29:30] This is our delta X right here.
[00:29:34] And this is our actual change in Y.
[00:29:37] Delta Y.
[00:29:40] Okay.
[00:29:41] Does this make sense so far?
[00:29:43] Okay.
[00:29:44] We have two different different directions.
[00:29:45] One is called dy and the other is called dx.
[00:29:56] This delta X is the same as our DX.
[00:30:03] Our differential DY is this distance right here.
[00:30:13] DY.
[00:30:15] It's the change from our equation of the tangent line as opposed to just the actual change in the function.
[00:30:27] That's what the differentials are.
[00:30:30] You don't really need to know that.
[00:30:33] Okay.
[00:30:33] But this was just a brief explanation.
[00:30:36] You don't have to memorize that.
[00:30:38] But that's what they are.
[00:30:43] And it's very simple.
[00:30:46] dy over dx equals f prime of x.
[00:30:52] Those mean the same thing, right?
[00:30:55] Yes.
[00:30:59] This is not a fraction even though it looks like a fraction, right?
[00:31:04] What is this?
[00:31:05] This is the derivative of this function Y with respect to X.
[00:31:10] But all you have to do is pretend it is a fraction and you multiply.
[00:31:16] And that's not really what's happening.
[00:31:18] Multiply both sides by DX.
[00:31:22] It looks like these cancel.
[00:31:26] And we get this.
[00:31:32] And this is it.
[00:31:33] Our differential dy is equal to the derivative times the differential dx.
[00:31:49] And we just have to know this.
[00:31:52] This is going to be very important later.
[00:31:55] Okay.
[00:31:55] So quick example.
[00:32:04] Sine of x.
[00:32:06] Find the differential dy.
[00:32:14] All we do is we take the derivative, which is cosine of X, right?
[00:32:25] And the derivative is the same thing as DY DX.
[00:32:29] We set it equal to cosine of x.
[00:32:35] Just pretend that you're multiplying by DX.
[00:32:41] And we get our differentials.
[00:32:42] dy is cosine of x times dx.
[00:32:51] dy is the derivative of the function times our differential dx.
[00:32:58] That's all you got to know how to do.
[00:33:01] We could find this DY here for that function we just did.
[00:33:08] dy for this function is cosine of x times dx.
[00:33:16] And that will give you the change right there.
[00:33:21] You don't really need that.
[00:33:22] All you got to do is you find the derivative, which is DY DX, multiplied by DX, even though that's not what's happening.
[00:33:32] And that's your answer.
[00:33:34] Okay.
[00:33:36] What if I want to find DX?
[00:33:38] What is it?
[00:33:44] We divide both sides by cosine.
[00:33:48] One over cosine times DY.
[00:33:56] And sometimes we write it like this.
[00:33:58] dy over cosine of x.
[00:34:02] Okay.
[00:34:02] This is going to be very important later.
[00:34:05] All you need to know is dy equals the derivative times dx.
[00:34:10] Okay.