Taylor Polynomials
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 6.3: “Taylor and Maclaurin Series” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series |
| Textbook used in class | Stewart, Calculus, Section 11.10: “Taylor and Maclaurin Series” (Examples 1, 3, 5) |
Opening Scenario
A Taylor polynomial is a finite version of a Taylor series: you stop adding terms at some degree $n$ and use the resulting polynomial to approximate $f$ near the center $a$. Calculators use this idea; to evaluate $e^{1.5}$, a device computes partial sums of the $e^x$ series until the answer is accurate enough. The key question is always how far the polynomial strays from the true function, which is answered by the remainder.
Quick Reference
$n$-th degree Taylor polynomial of $f$ at $a$: \[ T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n. \]
Remainder: $R_n(x) = f(x) - T_n(x)$.
Key property: $T_n$ matches $f$ and all its derivatives through order $n$ at $x = a$: \[ T_n^{(k)}(a) = f^{(k)}(a), \quad k = 0, 1, \ldots, n. \]
$T_n$ is the unique polynomial of degree $\leq n$ with this property.
Key Concepts
1. Building $T_n$ Step by Step
Example 1. Find the Taylor polynomials $T_1, T_2, T_3, T_4$ for $f(x) = e^x$ at $a = 0$.
$f^{(n)}(0) = 1$ for all $n$.
\[ T_1(x) = 1 + x, \quad T_2(x) = 1 + x + \frac{x^2}{2!}, \quad T_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}, \quad \ldots \]
Each $T_n$ is the previous one with one more term added. As $n$ grows, $T_n(x) \to e^x$ for every $x$.
2. Polynomials for $\sin$ and $\cos$
Example 2. Write the degree-5 Taylor polynomial for $\sin x$ at $a = 0$.
Only odd-degree coefficients are nonzero: \[ T_5(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} = x - \frac{x^3}{6} + \frac{x^5}{120}. \]
For small $|x|$, $T_1(x) = x$ gives a rough approximation, $T_3$ is better, and $T_5$ is even better. The exact error is controlled by $R_5(x)$.
3. Using $T_n$ as an Approximation
$T_n(x)$ is easy to evaluate on a calculator. The approximation is most accurate when $x$ is close to the center $a$.
Example 3. Use $T_3$ for $e^x$ to approximate $e^{0.1}$.
$T_3(0.1) = 1 + 0.1 + (0.1)^2/2 + (0.1)^3/6 = 1 + 0.1 + 0.005 + 0.000167 \approx 1.10517$.
The true value is $e^{0.1} \approx 1.10517$; the error is less than $10^{-7}$.
thinking higher degree always gives a better approximation everywhere. $T_n$ is accurate near $x = a$. Far from $a$, even a high-degree polynomial can diverge from $f$. A polynomial grows unboundedly as $|x| \to \infty$, while $\sin x$, for instance, stays bounded. The Taylor polynomial is a local approximation, not a global one.
Common Errors Summary
| Error | Correction |
|---|---|
| Including too many terms (forgetting where to stop) | $T_n$ has degree at most $n$; stop at $k = n$ |
| Writing $f^{(k)}(a)/k$ instead of $f^{(k)}(a)/k!$ | The coefficient is divided by $k!$, not $k$ |
| Confusing $T_n$ with the Taylor series | $T_n$ is a finite polynomial; the series is an infinite sum; $f(x) = T_n(x) + R_n(x)$ |
Leveled Practice
Problem 1. Find $T_4(x)$ for $\cos x$ at $a = 0$.
Show answer
Only even derivatives survive at $0$: $f(0) = 1$, $f''(0) = -1$, $f^{(4)}(0) = 1$.
$T_4(x) = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} = 1 - \dfrac{x^2}{2} + \dfrac{x^4}{24}$.
Problem 2. Approximate $\ln(1.2)$ using $T_3$ for $\ln(1+x)$ at $a = 0$.
Show answer
$T_3(x) = x - x^2/2 + x^3/3$. At $x = 0.2$: $T_3(0.2) = 0.2 - 0.02 + 0.00267 \approx 0.18267$.
True value: $\ln(1.2) \approx 0.18232$. Error $\approx 0.0004$.
Mastery Checklist
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