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Linearization: Concept and Example

Q:  Find the linearization of f(x) = ex at x = 0

A:  First, some concept:  Linearization is the act of finding a “linear function” that can approximate the given function on or around a given point.  In this problem, we want to find a line that models the shape of ex when you are around the point x = 0.

Step 1:  Find an (x, y) point on the function in question.

The function is f(x) = ex .  The x part of the point is 0.  Plug that in to find y:

f(0) = e0 = 1

So, the point is (0, 1).  Hold this point.  We will need it for later.

Essentially, we want to find a line that follows that patterns of f(x) = ex and goes through the point (0, 1).

So, to model the pattern, we need to slope of f(x) = ex at the given point.

Step 2:  Find the derivative (slope) of the function at the given point:

f(x) = ex

Find the derivative:

f ‘ (x) = ex

Now, find the slope at (0, 1):

f ‘ (0) = e0 = 1

So, the slope is 1.  m = 1

Step 3:  Find a line that has the same slope as the function that goes through the given point:

We need a line with slope of 1 that goes through the point (0, 1).

Start with your equation of a line:

y = mx + b

We know m = 1, so:

y = x + b

Plug in the (x, y) point to find “b”

1 = 0 + b

1 = b

So, the equation of the line is y = x + 1

In summation, to approximate values of f(x) = ex around where x = 0 you can use the line y = x + 1