Taylor and Maclaurin Series

Introduction

Taylor and Maclaurin series are powerful mathematical tools that allow us to approximate functions using polynomials. These series are particularly useful in physics when we need to simplify complex functions or analyze the behavior of systems near a specific point. In this section, we'll introduce both series and demonstrate their applications with several examples.

The Maclaurin Series

The Maclaurin series is a special case of the Taylor series, where the expansion is centered at $x = 0$. If a function $f(x)$ has derivatives of all orders at $x = 0$, then it can be expressed as an infinite series:

where $f^{(n)}(0)=\frac{d^n f}{dx^n}\bigg|_{x=0}$ denotes the $n$-th derivative of $f$ evaluated at $x = 0$, and $n!$ is the factorial of $n$.

The Maclaurin series provides a polynomial approximation of the function near $x = 0$. For many functions, this series converges to the function value within a certain radius of convergence.

The Taylor Series

The Taylor series is a more general expansion that allows us to approximate a function around any point $a$. If a function $f(x)$ has derivatives of all orders at $x = a$, then it can be expressed as:

where $f^{(n)}(a)=\frac{d^n f}{dx^n}\bigg|_{x=a}$ is the $n$-th derivative of $f$ evaluated at $x = a$. Notice that when $a = 0$, the Taylor series reduces to the Maclaurin series.

The Taylor series is useful when we want to approximate a function near a specific point other than the origin.

Example: Maclaurin Series for $\sin(x)$

Let's derive the Maclaurin series for $f(x) = \sin(x)$. We need to compute the derivatives of $\sin(x)$ and evaluate them at $x = 0$:

• $f(0) = \sin(0) = 0$
• $f'(x) = \cos(x)$, so $f'(0) = \cos(0) = 1$
• $f''(x) = -\sin(x)$, so $f''(0) = -\sin(0) = 0$
• $f'''(x) = -\cos(x)$, so $f'''(0) = -\cos(0) = -1$
• $f^{(4)}(x) = \sin(x)$, so $f^{(4)}(0) = \sin(0) = 0$

The pattern repeats every four derivatives. Substituting into the Maclaurin series formula:

This series converges for all real values of $x$. Notice that only odd powers of $x$ appear, which reflects the fact that $\sin(x)$ is an odd function: $\sin(-x) = -\sin(x)$.

Example: Maclaurin Series for $\cos(x)$

For $f(x) = \cos(x)$, we compute the derivatives:

• $f(0) = \cos(0) = 1$
• $f'(x) = -\sin(x)$, so $f'(0) = -\sin(0) = 0$
• $f''(x) = -\cos(x)$, so $f''(0) = -\cos(0) = -1$
• $f'''(x) = \sin(x)$, so $f'''(0) = \sin(0) = 0$
• $f^{(4)}(x) = \cos(x)$, so $f^{(4)}(0) = \cos(0) = 1$

The pattern also repeats every four derivatives. The Maclaurin series is:

This series also converges for all real $x$. Only even powers of $x$ appear, reflecting that $\cos(x)$ is an even function: $\cos(-x) = \cos(x)$.

Example: Maclaurin Series for $e^x$

For the exponential function $f(x) = e^x$, all derivatives are equal to $e^x$:

• $f(0) = e^0 = 1$
• $f'(0) = e^0 = 1$
• $f''(0) = e^0 = 1$
• $f^{(n)}(0) = 1$ for all $n$

Substituting into the Maclaurin series:

This series converges for all real $x$ and is one of the most important series in mathematics. It provides a way to compute $e^x$ for any value of $x$, and it's the basis for many other important mathematical results.

Common Taylor Series Approximations (Three Non-Zero Terms)

The following table provides three-term Taylor series approximations for common mathematical functions. These approximations are valid for small values of $x$ (or small values of the deviation from the expansion point). The table shows the first three non-zero terms in each expansion.

Note: For trigonometric functions, $x$ must be in radians. The three-term approximations are more accurate than the two-term versions. For example, $\sin(x) \approx x - x^3/6$ is accurate to within 1% for $|x| < 0.5$ radians (about 29°), compared to $|x| < 0.24$ radians for the one-term approximation.

These approximations are particularly useful in physics when analyzing systems near equilibrium or when making small-angle approximations. They allow us to simplify complex expressions and gain physical insight into the behavior of systems.

Application: Estimating Projectile Range Near 45°

As we saw in the projectile motion section, the range of a projectile launched with initial speed $v_0$ at angle $\theta$ is:

The maximum range occurs at $\theta = 45°$ (or $\theta = \pi/4$ radians). Suppose we want to estimate how the range changes when the launch angle is slightly displaced from $45°$. Let $\theta = \pi/4 + \delta$, where $\delta$ is a small angle deviation.

We can use a Taylor series expansion around $\theta = \pi/4$ to approximate $R(\theta)$:

First, let's compute the derivatives. Using $R(\theta) = \frac{v_0^2}{g}\sin(2\theta)$:

Evaluating at $\theta = \pi/4$:

• $R(\pi/4) = \frac{v_0^2}{g}\sin(\pi/2) = \frac{v_0^2}{g}$
• $R'(\pi/4) = \frac{2v_0^2}{g}\cos(\pi/2) = 0$
• $R''(\pi/4) = -\frac{4v_0^2}{g}\sin(\pi/2) = -\frac{4v_0^2}{g}$

Substituting into the Taylor series:

Notice that the first-order term (linear in $\delta$) is zero! This confirms that $\theta = 45°$ is indeed a maximum. The range decreases quadratically with the deviation $\delta$ from the optimal angle.

For small deviations, we can approximate:

This tells us that small deviations from $45°$ have a relatively small effect on the range. For example, if $\delta = 5° = \pi/36 \approx 0.087$ radians, then:

So a $5°$ deviation from the optimal angle reduces the range by only about $1.5\%$. This demonstrates why the range is relatively insensitive to small changes in launch angle near the optimum.