# Step By Step Calculus » 13.1 - Linear Approximation and Newton's Method

Synopsis
Suppose a\in \mathbb{R}a\in \mathbb{R} is some fixed number, which we will call an anchor, and we think of xx as varying close to aa. Then, we know from the mean value theorem that
\displaystyle f(x)=f(a)+f^{\prime }(y)(x-a)
\displaystyle f(x)=f(a)+f^{\prime }(y)(x-a)
for some yy between xx and aa that can depend upon xx. Now, if f^{\prime }f^{\prime } is continuous at aa and xx is close to aa, then it is reasonable to expect f^{\prime }(y)f^{\prime }(y) is close to f^{\prime }(a)f^{\prime }(a) and
\displaystyle f(x)\approx f(a)+f^{\prime }(a)(x-a).
\displaystyle f(x)\approx f(a)+f^{\prime }(a)(x-a).
This gives us our linear approximation
\displaystyle L_{a}(x)=f(a)+f^{\prime }(a)(x-a)
\displaystyle L_{a}(x)=f(a)+f^{\prime }(a)(x-a)