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What are Taylor and Maclaurin polynomials?
in Mathematics by Platinum (141,884 points) | 15 views

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(Taylor Polynomial, Maclaurin Polynomial). Let \(f\) be a function whose first \(n\) derivatives exist at \(x=c\).
1. The Taylor polynomial of degree \(n\) of \(f\) centered at \(x=c\) is
\[
p_n(x)=f(c)+f^{\prime}(c)(x-c)+\frac{f^{\prime \prime}(c)}{2 !}(x-c)^2+\frac{f^{\prime \prime \prime}(c)}{3 !}(x-c)^3+\cdots+\frac{f^{(n)}(c)}{n !}(x-c)^n .
\]
2. A special case of the Taylor polynomial is the Maclaurin polynomial, where \(c=0\). That is, the Maclaurin polynomial of degree \(n\) of \(f\) is
\[
p_n(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^2+\frac{f^{\prime \prime \prime}(0)}{3 !} x^3+\cdots+\frac{f^{(n)}(0)}{n !} x^n .
\]
by Platinum (141,884 points)

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