Q: Find the linearization of f(x) = e^{x} 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 e^{x} 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) = e^{x} . The x part of the point is 0. Plug that in to find y:

f(0) = e^{0} = 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) = e^{x} and goes through the point (0, 1).

So, to model the pattern, we need to slope of f(x) = e^{x} at the given point.

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

f(x) = e^{x}

Find the derivative:

f ‘ (x) = e^{x}

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

f ‘ (0) = e^{0} = 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) = e^{x} around where x = 0 you can use the line **y = x + 1**