Linear Approximation in Calculus: Formula & Examples

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  • 0:03 Linnear…
  • 0:47 Tangent Lines and…
  • 1:38 Formula for Linearization
  • 2:53 Using Linearization
  • 4:57 Lesson Summary
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Lesson Transcript
Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

Expert Contributor
Alfred Mulzet

Dr. Alfred Kenric Mulzet received his Ph.D. in Applied Mathematics from Virginia Tech. He currently teaches at Florida State College in Jacksonville.

Linear approximation is a method for estimating a value of a function near a given point using calculus. In this lesson, you'll learn how to find a linear approximation and see an example of how it can be used.

Linear Approximation/Linearization

Linear approximation is a method of estimating the value of a function, f(x), near a point, x = a, using the following formula:

Linearization Formula

The formula we're looking at is known as the linearization of f at x = a, but this formula is identical to the equation of the tangent line to f at x = a.

This lesson shows how to find a linearization of a function and how to use it to make a linear approximation. This method is used quite often in many fields of science, and it requires knowing a bit about calculus, specifically, how to find a derivative.

Tangent Lines and Linearization

Let's review a basic fact about derivatives. The value of the derivative at a specific point, x = a, measures the slope of the curve, y = f(x), at that point. In other words, f '(a) = slope of the tangent line at a.

The tangent line to a function at a specific point.
Linear approximation as a tangent line

Now, the tangent line is special because it's the one line that matches the direction of the curve most closely, at the specific x-value you are interested in. Notice how close the y-values of the function and the tangent line are when x is near the point where the tangent line meets the curve.

So, if the curve y = f(x) is way too complicated to work with, and if you're only interested in values of the function near a particular point, then you could throw away the function and just use the tangent line. Well, don't actually throw away the function. . . we may need it later!

Formula for Linearization

So, how do you find the linearization of a function f at a point x = a? Remember that the equation of a line can be determined if you know two things:

  1. The slope of the line, m
  2. Any single point that the line goes through, (a, b).

We plug these pieces of info into the point-slope form, and this gives us the equation of the line. (This is just algebra, folks; no calculus yet.)

y - b = m(x-a)

But, in problems like these, you will not be given values for b or m. Instead, you have to find them yourself. Firstly m = f '(a), because the derivative measures the slope, and secondly, b = f(a), because the original function measures y-values.

Putting it all together and solving for y:

Formula for linearization of a function

The last line is precisely the linearization of the function f at the point x = a. Now that we know where the formula comes from, let's use it to find a linear approximation.

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Additional Activities

Linear Approximation:

The method of linear approximation can be used for a variety of functions. Linear approximation is sometimes used to estimate the value of some function. For example, we can use linear approximation for a natural log function. Remember that the derivative of f(x) = ln x is f'(x) = 1/x.


Find the approximate value of ln 1.1 and ln 0.9 using linear approximation.


We will use the function f(x) = ln x, and the value a = 1. Then

f'(x) = 1/x.

f'(a) = f'(1) = 1.

Furthermore, f(a) = ln 1 = 0.

Therefore, the linear approximation of f at x = 1 is

y = f(1) + f'(1) (x - 1)

y = 0 + 1 (x - 1)

y = x - 1

The linear approximation gives us the following values:

f(1.1) = ln 1.1 can be approximated by

y = 1.1 - 1 = 0.1.

f(0.9) can be approximated by

y = 0.9 - 1 = -0.1.

Now, a calculator shows us that ln 1.1 is approximately 0.09531 and ln 0.9 approximately -1054.

These are fairly accurate approximations.

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