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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.

Answer and Explanation:

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}


Learn more about this topic:

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Linear Approximation in Calculus: Formula & Examples

from College Preparatory Mathematics: Help and Review

Chapter 6 / Lesson 8
151K

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