A Taylor polynomial is a partial sum of the Taylor series. It provides a polynomial approximation of a function near the center point $a$. The higher the degree $n$, the better the approximation (within the radius of convergence). Taylor polynomials are the foundation for numerical approximation methods in calculus.
🎯 In this section you will learn
- How to construct Taylor polynomials of degree $n$
- How to use Taylor polynomials to approximate function values
- How the approximation improves as $n$ increases
- How to find the Maclaurin polynomial (center at $0$)
📌 Taylor Polynomial Formula
$$ P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n $$
Taylor polynomial of degree $n$ centered at $x=a$
💡 Maclaurin Polynomial
When $a=0$, the polynomial is called a Maclaurin polynomial: $P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k$.
Find the 3rd degree Maclaurin polynomial for $f(x) = e^x$.
①
Find derivatives at $0$
$f(0)=1$, $f'(0)=1$, $f''(0)=1$, $f'''(0)=1$
②
Apply the formula
$P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$
Find the 5th degree Maclaurin polynomial for $f(x) = \sin x$.
- $\sin 0 = 0$, $\sin' 0 = 1$, $\sin'' 0 = 0$, $\sin''' 0 = -1$, $\sin^{(4)} 0 = 0$, $\sin^{(5)} 0 = 1$
- $P_5(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} = x - \frac{x^3}{6} + \frac{x^5}{120}$
Use the 3rd degree Maclaurin polynomial for $e^x$ to approximate $e^{0.5}$.
①
$P_3(0.5) = 1 + 0.5 + \frac{(0.5)^2}{2} + \frac{(0.5)^3}{6}$
$= 1 + 0.5 + 0.125 + 0.0208333 = 1.6458333$
②
Compare with actual
$e^{0.5} \approx 1.648721$ — error about $0.0029$
Find the 2nd degree Taylor polynomial for $f(x) = \ln x$ centered at $a=1$.
①
Find derivatives at $1$
$f(1)=0$, $f'(1)=1$, $f''(1)=-1$
②
Apply the formula
$P_2(x) = 0 + 1(x-1) + \frac{-1}{2!}(x-1)^2 = (x-1) - \frac{(x-1)^2}{2}$
③
Approximate $\ln(1.2)$
$P_2(1.2) = (0.2) - \frac{(0.2)^2}{2} = 0.2 - 0.02 = 0.18$
Actual $\ln(1.2) \approx 0.1823$
⚠️ Important Notes
- Higher degree = better approximation (within the interval of convergence).
- Taylor polynomials are most accurate near the center $a$. As you move away, the error grows.
- Even-degree polynomials may give better accuracy for symmetric functions like $\cos x$.
🔍 Key Takeaways
- Taylor polynomial: $P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k$.
- Maclaurin polynomial: Taylor polynomial with $a=0$.
- Use Taylor polynomials to approximate function values near the center.
- The approximation improves as $n$ increases (error decreases).
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