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What are Taylor and Maclaurin polynomials?
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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
$p_n(x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^2+\frac{f^{\prime \prime \prime}(0)}{3 !} x^3+\cdots+\frac{f^{(n)}(0)}{n !} x^n .$
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