Finding Taylor polynomial approximations of functions — AP Calculus BC
1. Definition of a Taylor Polynomial ★★☆☆☆ ⏱ 10 min
T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x-a)^k
2. Step-by-Step Calculation of Taylor Polynomials ★★★☆☆ ⏱ 15 min
To avoid common mistakes, follow this consistent process to build any Taylor polynomial:
- Compute all derivatives of $f(x)$ up to the $n$th derivative
- Evaluate each derivative at the center point $x=a$
- Divide each evaluated derivative by the corresponding factorial $k!$ for the $k$th term
- Multiply each term by $(x-a)^k$ and add all terms together
3. Common Maclaurin Polynomials ★★☆☆☆ ⏱ 10 min
Maclaurin polynomials (Taylor at $a=0$) for basic functions appear constantly on the AP exam. You can use these standard forms to quickly construct new polynomials for related functions by substitution, instead of recalculating derivatives from scratch.
Common Pitfalls
Why: The Taylor formula requires division by factorial to match the derivative of the polynomial. Skipping this step gives a coefficient that is off by a factor of $k!$.
Why: Confusing Taylor and Maclaurin polynomials leads to an approximation that is only accurate near 0, not near the required center.
Why: Signs flip every time you take a derivative of $\sin x$, $\cos x$, or negative powers of $x$, so it is easy to end up with the wrong sign.
Why: Since counting starts at $k=0$, you need $n+1$ total terms to get an nth-degree polynomial.
Why: For example, substituting $3x$ into $\sin x$ often leads to the incorrect term $\frac{3x^3}{3!}$ instead of $\frac{(3x)^3}{3!}$.