Taylor and Maclaurin Polynomials and Series: Theory, Examples, and Applications

Taylor and Maclaurin Polynomials

Linear Approximation and Tangent Line

The linear approximation states that if a function f(x) is differentiable at a point x = a, then near x = a it can be approximated by its tangent line:

• Tangent line equation: $y = f(a) + f'(a)(x - a)$

• This is a first-order approximation, using only the value and first derivative at a.

Key Point: The tangent line provides a simple local approximation, but higher accuracy can be achieved using polynomials of higher degree.

Taylor Polynomial: Definition and Construction

A Taylor polynomial of degree n for f(x) at x = a is a polynomial p_n(x) that matches f(x) and its first n derivatives at x = a:

• $p_n(a) = f(a)$

• $p_n'(a) = f'(a)$

• $p_n''(a) = f''(a)$

• ... up to $p_n^{(n)}(a) = f^{(n)}(a)$

The general form is:

• $p_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$

• In sigma notation: $p_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!}(x-a)^k$

Key Point: The Taylor polynomial provides a polynomial approximation to f(x) near x = a, matching derivatives up to order n.

Maclaurin Polynomial: Special Case

The Maclaurin polynomial is a Taylor polynomial centered at a = 0:

• $p_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$

Key Point: Maclaurin polynomials are useful for approximating functions near the origin.

Examples of Taylor and Maclaurin Polynomials

• Example 1: For $f(x) = e^x$, the Maclaurin polynomial of degree n is $p_n(x) = 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$

• Example 2: For $f(x) = e^{-x}$, the Maclaurin polynomial of degree n is $p_n(x) = 1 - x + \frac{x^2}{2!} - \cdots + (-1)^n \frac{x^n}{n!}$

• Example 3: For $f(x) = \cosh(x) = \frac{e^x + e^{-x}}{2}$, the Maclaurin polynomial contains only even powers of x.

• Example 4: For $f(x) = \sinh(x) = \frac{e^x - e^{-x}}{2}$, the Maclaurin polynomial contains only odd powers of x.

• Example 5: For $f(x) = e^x$, the Taylor polynomial of degree n at x = 2 is $p_n(x) = e^2 + e^2(x-2) + \frac{e^2}{2!}(x-2)^2 + \cdots + \frac{e^2}{n!}(x-2)^n$

Key Point: The form of the polynomial depends on the function and the center a.

Even and Odd Functions in Maclaurin Series

• If f(x) is even ($f(-x) = f(x)$), its Maclaurin polynomial contains only even powers of x.

• If f(x) is odd ($f(-x) = -f(x)$), its Maclaurin polynomial contains only odd powers of x.

• Examples: $\cos(x)$ is even, $\sin(x)$ is odd.

Maclaurin Polynomials for Sine and Cosine

By repeated differentiation, we find the derivatives cycle every four steps:

k	$f^{(k)}(x)$	$f^{(k)}(0)$
0	$\sin(x)$	0
1	$\cos(x)$	1
2	$-\sin(x)$	0
3	$-\cos(x)$	-1
4	$\sin(x)$	0

Thus, the Maclaurin polynomial for $\sin(x)$ (degree n even) is:

• $p_n(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots + \frac{x^n}{n!}$

For $\cos(x)$:

• $p_n(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots + \frac{x^n}{n!}$

Taylor's Theorem

Statement and Remainder Term

Taylor's Theorem provides a formula for the error (remainder) when approximating f(x) by its Taylor polynomial:

• Let $p_n(x)$ be the Taylor polynomial of degree n for f(x) at x = a. Then:

$R_n(x) = f(x) - p_n(x) = \int_a^x \frac{f^{(n+1)}(t)}{n!}(x-t)^n dt$

Key Point: If $R_n(x)$ is small, the Taylor polynomial is a good approximation to f(x).

Applications and Approximations

• Taylor and Maclaurin polynomials are used to approximate function values, especially when direct computation is difficult.

• Example: Approximating $\sin(\pi)$ using the Maclaurin polynomial for $\sin(x)$ is not accurate, since $\pi$ is far from 0. Better results are obtained when x is close to the center a.

• For $\sin(\frac{\pi}{6})$, the Taylor polynomial centered at $x = \frac{\pi}{6}$ gives a good approximation.

k	$f^{(k)}(x)$	$f^{(k)}(\frac{\pi}{6})$
0	$\sin(x)$	$\frac{1}{2}$
1	$\cos(x)$	$\frac{\sqrt{3}}{2}$
2	$-\sin(x)$	$-\frac{1}{2}$
3	$-\cos(x)$	$-\frac{\sqrt{3}}{2}$
4	$\sin(x)$	$\frac{1}{2}$

The Taylor polynomial of degree 4 for $\sin(x)$ at $x = \frac{\pi}{6}$ is:

• $p_4(x) = \frac{1}{2} + \frac{\sqrt{3}}{2}(x - \frac{\pi}{6}) - \frac{1}{4}(x - \frac{\pi}{6})^2 - \frac{\sqrt{3}}{12}(x - \frac{\pi}{6})^3 + \frac{1}{48}(x - \frac{\pi}{6})^4$

Degree of Approximation and Error Analysis

• The degree n required for a good approximation depends on how close x is to a and the behavior of higher derivatives.

• Higher-degree polynomials provide better approximations, but may require more computation.

Example: Limit Calculation Using Taylor Polynomials

• Limits involving complicated functions can be simplified using Taylor expansions.

• Example: $\lim_{x \to 0} \frac{6x^2 - \sin(2x) \sin(3x)}{x^4}$ can be evaluated by expanding $\sin(2x)$ and $\sin(3x)$ in Taylor series.

Summary Table: Taylor and Maclaurin Polynomials for Common Functions

Function	Maclaurin Polynomial (degree n)	Properties
$e^x$	$1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}$	All powers of x
$e^{-x}$	$1 - x + \frac{x^2}{2!} - \cdots + (-1)^n \frac{x^n}{n!}$	Alternating signs
$\sin(x)$	$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$	Odd powers only
$\cos(x)$	$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$	Even powers only
$\cosh(x)$	$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$	Even powers only
$\sinh(x)$	$x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$	Odd powers only

Additional info:

• Taylor and Maclaurin polynomials are foundational tools in calculus for approximating functions, analyzing limits, and understanding the behavior of functions near a point.

• The error term in Taylor's theorem provides a way to estimate the accuracy of the polynomial approximation.

• These polynomials are also used in numerical methods, such as Newton's method for finding roots.