Recover polynomial coefficients from derivatives at zero.
Notes and key methods
A Maclaurin expansion starts \(f(x)=f(0)+f'(0)x+f''(0)x^2/2!+f^{(3)}(0)x^3/3!+\cdots\). Each coefficient includes its factorial denominator.
For a product, multiply truncated series while collecting every contribution to a given power. A truncated expression is an approximation away from zero; state the retained order rather than treating it as an exact identity.
Check the conditions
The required derivatives exist near zero.
State the order to which the series is truncated.
Three worked examples
Example 1 · Maclaurin expansions
Let \(f(x)=e^{4 x} \cos{\left(x \right)}\). Write \(f(x)=1+Ax+Bx^2+Cx^3+O(x^4)\).
Find A.
Find B.
Find C.
Use the cubic polynomial to approximate f(\frac{1}{50}).