Maclaurin polynomials

Definition

Maclaurin polynomials are special cases of Taylor polynomials centered at $x = 0$. They provide polynomial approximations of functions using derivatives evaluated at zero.

5 Must Know Facts For Your Next Test

1. A Maclaurin polynomial is a Taylor polynomial centered at $x = 0$.
2. The general form of a Maclaurin polynomial for a function $f(x)$ is $P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$.
3. Maclaurin polynomials can approximate functions near zero with increasing accuracy as the degree of the polynomial increases.
4. The error in approximation by a Maclaurin polynomial can be analyzed using the remainder term in Taylor's theorem.
5. $e^x$, $\sin(x)$, and $\cos(x)$ have well-known Maclaurin series expansions.

Review Questions

• What is the difference between a Taylor polynomial and a Maclaurin polynomial?
• Write the general form of a Maclaurin polynomial for any function $f(x)$. What does each term represent?
• How does increasing the degree of a Maclaurin polynomial affect its approximation accuracy?

Related terms

Taylor Series: An infinite sum representing a function as an expansion about any point $a$, given by $f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$.

Power Series: An infinite series in the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where each term involves powers of $(x - a)$.

Convergence Radius: The distance from the center point within which a power series converges to the function it represents.