Taylor's theorem explains how a smooth function can be approximated near a chosen point by a polynomial built from the function's derivatives at that point. This matters because polynomials are often easier to compute, graph, and analyze than complicated functions. The theorem also tells us that the approximation is not just a guess, because it includes a remainder term that measures the error.
As you move farther from the expansion point, the Taylor polynomial may drift away from the true function, so understanding the remainder is essential.
Key Facts
- Taylor polynomial of degree n about a: P_n(x) = sum from k = 0 to n of f^(k)(a)(x - a)^k / k!
- Taylor's theorem: f(x) = P_n(x) + R_n(x), where R_n(x) is the remainder or error.
- Lagrange remainder form: R_n(x) = f^(n+1)(c)(x - a)^(n+1) / (n+1)! for some c between a and x.
- Lagrange error bound: |R_n(x)| <= M|x - a|^(n+1) / (n+1)! if |f^(n+1)(t)| <= M between a and x.
- The approximation is usually most accurate near the expansion point a because powers of |x - a| are small.
- A Taylor series represents f(x) exactly on an interval only if the remainder R_n(x) approaches 0 as n approaches infinity.
Vocabulary
- Taylor polynomial
- A polynomial that matches a function and its first several derivatives at a chosen expansion point.
- Expansion point
- The value a where the function's derivatives are used to build the Taylor polynomial.
- Remainder
- The difference R_n(x) = f(x) - P_n(x) between the true function value and the Taylor polynomial value.
- Lagrange error bound
- An inequality that gives a guaranteed maximum possible size for the Taylor approximation error.
- Radius of convergence
- The distance from the expansion point within which a Taylor series converges to a finite value.
Common Mistakes to Avoid
- Forgetting the factorial in each Taylor term is wrong because the k! in the denominator comes from repeated differentiation and changes every coefficient.
- Using derivatives at x instead of at the expansion point a is wrong because a Taylor polynomial is built from fixed derivative values evaluated at a.
- Treating the error bound as the exact error is wrong because the Lagrange bound gives a maximum possible error, not necessarily the actual difference.
- Assuming a Taylor polynomial works equally well far from a is wrong because the error often grows with powers of |x - a| and the series may not converge everywhere.
Practice Questions
- 1 Find the degree 3 Taylor polynomial for f(x) = e^x about a = 0, then use it to approximate e^0.2.
- 2 Use the Lagrange error bound to estimate the maximum error when approximating sin(0.3) by its degree 3 Taylor polynomial about a = 0.
- 3 Explain why a Taylor polynomial centered at a = 0 might approximate ln(1 + x) well for x = 0.1 but poorly for x = 1.5.