# Find a linear approximation formula, L(x) , for f(x)=e^x near x=0 .

## Question:

Find a linear approximation formula, {eq}L(x) {/eq}, for {eq}f(x)=e^x {/eq} near {eq}x=0 {/eq}.

## Linear Approximation:

In mathematics specifically in numerical mathematics the linearization process refers to finding a linear approximation of a function at a certain point.

Among many applications in science, this linear approach is used to find the stability or equilibrium of some systems.

Calculate the values of the function and its first order derivative

{eq}\displaystyle f(x)=e^x \,\, \Longrightarrow \,\, f(0)=1\\ \displaystyle f'(x)=e^x \,\, \Longrightarrow \,\, f'(0)=1 {/eq}

Apply the formula for the linearization of the function at the given point

{eq}\displaystyle L(x)=f(a)+f'(a)(x-a) \,\, \Longrightarrow \,\, \textrm {linearization of the function f(x) at the point x=a}\\ \displaystyle L(x)=f(0)+f'(0)(x-0) \,\, \Longrightarrow \,\, \textrm {linearization of the function f(x) at the point x=0}\\ \displaystyle L(x)=1+x \,\, \Longrightarrow \,\, \textrm {linearization of the function} \,\,\,\, f(x)=e^x \,\,\,\, \textrm {at the point x=0} {/eq}

The linear approximation for the given function is: {eq}\boxed{L(x)=1+x} {/eq} 