Taylor polynomials let us approximate a complicated function with a polynomial built from information at one chosen center point. This matters because polynomials are easy to evaluate, graph, differentiate, and integrate. Near the center, a Taylor polynomial can closely match the original function, making it a powerful tool in calculus, physics, engineering, and computation.
The polynomial uses the function value and derivative values at x = a to copy the function's local behavior. A higher degree polynomial matches more derivatives at the center, so it usually follows the curve better near that point. The approximation often becomes worse farther from the center, and the size of the error depends on the function, the degree, and the distance from a.
Key Facts
- Taylor polynomial of degree n centered at a: P_n(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + ... + f^(n)(a)(x - a)^n/n!
- Linear approximation is the first degree Taylor polynomial: P_1(x) = f(a) + f'(a)(x - a).
- Quadratic approximation is P_2(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2.
- Maclaurin polynomials are Taylor polynomials centered at a = 0.
- Taylor polynomials match derivatives at the center: P_n^(k)(a) = f^(k)(a) for k = 0, 1, 2, ..., n.
- Lagrange error bound: |R_n(x)| <= M|x - a|^(n + 1)/(n + 1)! if |f^(n + 1)(t)| <= M between a and x.
Vocabulary
- Taylor polynomial
- A polynomial that approximates a function near a center point using the function's derivatives at that point.
- Center
- The value a where the Taylor polynomial is built and where it matches the function most closely.
- Degree
- The highest power of x - a included in the Taylor polynomial.
- Remainder
- The difference between the actual function value and the Taylor polynomial approximation.
- Maclaurin polynomial
- A Taylor polynomial centered at a = 0.
Common Mistakes to Avoid
- Forgetting the factorial in each term is wrong because the kth derivative term must be divided by k! to match the function's derivative values correctly.
- Using powers of x instead of x - a is wrong for a Taylor polynomial centered at a because the expansion measures distance from the center.
- Assuming higher degree is always better everywhere is wrong because Taylor polynomials usually improve the fit mainly near the center and can behave poorly far away.
- Mixing up f'(a), f''(a), and f^(k)(a) with f'(x), f''(x), and f^(k)(x) is wrong because the Taylor coefficients are derivative values evaluated at the center.
Practice Questions
- 1 Find the degree 2 Taylor polynomial for f(x) = e^x centered at a = 0, then use it to approximate e^0.2.
- 2 Find the degree 3 Taylor polynomial for f(x) = sin x centered at a = 0, then use it to approximate sin(0.5).
- 3 A graph shows f(x), P_1(x), P_2(x), and P_4(x) centered at x = a. Explain why all the curves are closest together near x = a and why they may separate as x moves farther from a.