Calculus II · Unit 4B · lesson

Taylor's Theorem and the Remainder

Concept

Learning objectives

use Taylor's theorem to express approximation error and identify a derivative bound.

Taylor's Theorem and the Remainder

Explanation

Approximation becomes mathematics only when error is controlled

A Taylor polynomial matches derivatives at the center, but matching alone does not say how far away the approximation remains accurate. Taylor's theorem represents the difference $f(x)-P_n(x)$ using the next derivative evaluated at an unknown intermediate point. Bounding that derivative produces a usable error guarantee.

The unknown point $c$ lies between the center and the target input. We do not need to find it; we need a bound $M$ valid throughout that interval. Then factorial growth in the denominator often drives the error rapidly toward zero. The theorem is the bridge from formal Taylor coefficients to a proven representation.

Bridge

Approximation becomes mathematics when the error is bounded

A Taylor polynomial may look accurate on a graph, but a graph cannot certify digits. Taylor's Theorem expresses the exact error after degree $n$ as

R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}

for some point $c$ between $a$ and $x$ .

We usually do not know $c$ . Instead, bound $|f^{(n+1)}(t)|\le M$ throughout the interval and obtain

|R_n(x)|\le \frac{M|x-a|^{n+1}}{(n+1)!}.

The factorial in the denominator explains why many Taylor approximations improve rapidly near the center.

**A Taylor polynomial comes with an error envelope**
A function and its Taylor polynomial meet at the center, while symmetric upper and lower error bounds widen with distance from the center.
A Taylor polynomial comes with an error envelope. Function, Taylor polynomial, and an error envelope widening away from the center.

Decision

Taylor error-bound checklist

Identify the center $a$ , degree $n$ , and target $x$ . Find a bound $M$ for $|f^{(n+1)}(t)|$ on the entire interval between $a$ and $x$ . Substitute into the Lagrange bound, preserve enough precision until the end, and compare the result with the requested tolerance.

Proof idea

The remainder theorem is a high-order Mean Value Theorem

The proof constructs an auxiliary function whose zeros force repeated applications of Rolle's Theorem. The resulting point $c$ converts unmatched higher-order behavior into the exact remainder formula. The full proof is longer, but the mechanism is the same idea that converts average change into a derivative value somewhere inside an interval.

**A remainder bound encloses the unknown error**
Taylor's theorem does not merely offer a polynomial; it gives an explicit envelope within which the true function must lie.
A remainder bound encloses the unknown error. Function, Taylor polynomial, and an error envelope widening away from the center.

How to read the visual

The dashed polynomial is the approximation. The shaded band represents a deliberately simple error allowance, and the true exponential curve stays inside it on the displayed interval.

Concept

Lagrange remainder

For some $c$ between $a$ and $x$ ,

R_n(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.

Hence if $|f^{(n+1)}(t)|\le M$ , then

|R_n(x)|\le\frac{M|x-a|^{n+1}}{(n+1)!}.

Guided walkthrough

Bound exponential error

Approximate $e^{0.5}$ with $P_3$ . On $[0,0.5]$ , the fourth derivative is $e^t\le e^{0.5}$ . Therefore

|R_3(0.5)|\le\frac{e^{0.5}(0.5)^4}{4!}<0.0043.

Worked example

Certify a sine approximation

Approximate $\sin(0.2)$ by $T_3(x)=x-x^3/6$ . The next derivative after degree three is $\sin x$ , whose absolute value is at most $1$ . Therefore

|R_3(0.2)|\le\frac{0.2^4}{4!} =\frac{0.0016}{24}<0.000067.

Thus the cubic approximation is guaranteed accurate to within about $6.7\times10^{-5}$ .

Common mistake

M must bound the derivative on the entire interval

Using only $|f^{(n+1)}(a)|$ is not generally enough. The bound must hold between the center and the target input.

Interactive checku4b-taylor_remainder_theorem-01

State the standard Lagrange upper bound for $|R_n(x)|$ .

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Show hint

Use a bound $M$ on the $(n+1)$ st derivative.

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Exercise

Bound the error in approximating $\sin(0.2)$ by $x-x^3/6$ .

Exercise

Find a derivative bound for $\ln x$ near $x=1$ .

Exercise

Explain why the intermediate point need not be known.

Exercise

Use a bound to prove the exponential Taylor series converges to $e^x$ .

After the explanation

Use the section idea

Reading lens

Build new expansions from a small verified library using explicit substitutions, products, derivatives, or integrals.

Mental model

A standard series is a reusable identity with a domain, not a formula fragment detached from convergence.

Decision

Name the source series, apply one transformation at a time, and transform its interval alongside it.

Common trap

Recognizing a pattern but omitting the substitution's effect on the interval.

Check yourself

Can you reconstruct the expansion and its domain without memorizing the final line?

Source & rights

Original instruction with traceable references.

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