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.
Understanding Calculus: Taylor's Theorem and the Remainder
A Taylor polynomial works by matching local behaviour one derivative at a time. The constant term matches the function value at the chosen centre. The linear term matches the slope there.
Higher-degree terms match bending, changes in bending, and further changes in shape. This is why the polynomial can closely follow a curve for a short distance.
The factorial in each coefficient is important. Repeated differentiation brings down multiplying factors from powers, and the factorial cancels those factors so that each derivative has exactly the required value at the centre.
The remainder is not usually found by first locating the mysterious point inside the interval. Instead, it is often used as a bound. Find a number that is at least as large as the absolute value of the next derivative throughout the part of the graph being used.
Then combine that number with the distance from the centre and the factorial of the next degree. This gives a worst-case error. A bound can be larger than the actual error, sometimes much larger, but it is reliable.
Care is needed when choosing the maximum derivative. It must work everywhere between the centre and the input value, not merely at one endpoint.
For example, near zero, a polynomial for sine begins with the input minus the input cubed divided by six. The next omitted term involves the input to the fifth power divided by one hundred twenty. Since every derivative of sine has absolute value no greater than one, the size of that next-order error is easy to control.
At an input close to zero, the fifth power is tiny. At an input farther away, it grows quickly.
This explains why a low-degree approximation can be excellent for a small angle in radians yet poor for a larger angle. Engineers use this idea when a calculator, sensor, or computer needs a fast estimate from simple arithmetic.
Students should separate three claims that are easy to mix up. A polynomial may match several derivatives at one point without being accurate far away. An error bound proves a maximum possible error, not the exact error.
A Taylor series may have terms that become small while still failing to equal the original function at every input. The limiting remainder decides that last issue. When solving problems, state the centre, the degree, and the interval clearly.
Check which derivative controls the remainder. Keep units sensible in applied work, since an error in an angle, length, or time can affect later calculations.
Finally, compare the bound with the accuracy required. A bound of one thousandth is useful only when that level of uncertainty is acceptable.
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.