Problem 7 - Essential Calculus

Question

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\sin 2 x, \quad c=0 $$

Step-by-Step Solution

Answer

The Taylor series for \(\sin(2x)\) centered at 0 is \[ \sum_{n=0}^{\infty} \frac{(-1)^n * (2x)^{2n+1}}{(2n+1)!} \]

Step 1: Identify the Function and Center

Identify the function as \(f(x) = \sin(2x)\) and the center of the Taylor series as \(c=0\).

Step 2: Calculate Derivatives

Calculate the first few derivatives of the function evaluated at 0. \n The first few derivatives are: \n\n \(f'(x) = 2\cos(2x)\) \nEvaluation at \(x=0\) gives \(f'(0) = 2\),\n\n \(f''(x) = -4\sin(2x)\) \nEvaluation at \(x=0\) gives \(f''(0) = 0\),\n\n \(f'''(x) = -8\cos(2x)\) \nEvaluation at \(x=0\) gives \(f'''(0)= -8\), \n\n \(f''''(x) = 16\sin(2x)\) \nEvaluation at \(x=0\) gives \(f''''(0)= 0\), and so on.

Step 3: Recognize Pattern

Recognize the pattern in the derivatives and their evaluations, the derivatives cycle every four steps. Rather than a random series of coefficients, they return to the original state over and over (2, 0, -8, 0, 32, 0, -128, 0, etc.).

Step 4: Formulate the Taylor Series

Formulate the Taylor series using the pattern observed. The general expression for Taylor series is: \(f(x) = \sum_{n=0}^{\infty}\frac{f^n(c) * (x - c)^n}{n!}\) \nSubstituting the observed pattern, the Taylor series for \(\sin(2x)\) centered at 0 becomes: \[ f(x) = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \frac{(2x)^7}{7!} + ... = \sum_{n=0}^{\infty} \frac{(-1)^n *(2x)^{2n+1}}{(2n+1)!} \]

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, describing how a function changes as its input changes. In the context of finding Taylor Series terms, derivatives help us determine the coefficients that will contribute to the series.

• The first derivative, denoted as \(f'(x)\), provides the slope of the tangent to the curve at a given point, showing the rate of change.
• The second derivative, \(f''(x)\), measures the rate at which the rate of change itself is changing, often related to concavity throughout the curve.
• Higher-order derivatives continue this trend, allowing us to capture more detail about the function's behavior.

For the function \(f(x) = \sin(2x)\), derivatives are calculated sequentially:

• First derivative: \(f'(0) = 2\)
• Second derivative: \(f''(0) = 0\)
• Third derivative: \(f'''(0) = -8\)
• Fourth derivative: \(f''''(0) = 0\)

This pattern repeats itself every four terms, indicating an oscillating nature inherent to trigonometric functions.

Trigonometric Functions

Trigonometric functions like \( \sin(x) \) and \( \cos(x) \) are essential in both pure and applied mathematics, often used to model periodic phenomena like waves. For applying Taylor series, we focus on the function's behavior around a specific point, known as the center, denoted \(c\).

• \(\sin(2x)\) is a scaled version of the basic sine function, meaning all its properties like amplitude and frequency are doubled.
• These functions are naturally periodic representations of repeating cycles, which directly impacts their Taylor series.

The periodic nature of trigonometric functions manifests in regular cycling of their derivatives, leading to repeating patterns in the Taylor expansion. This characteristic ensures expansions of trigonometric functions often converge quickly, making them efficient to work with in computations.

Series Expansion

Series expansion allows complex functions to be expressed as the sum of a series of terms, simplifying analysis and calculations. Taylor series, in particular, represent functions as polynomials, providing easy-to-use approximations for small intervals around a central value \(c=0\).

• The general Taylor series formula is \(f(x) = \sum_{n=0}^{\infty}\frac{f^n(c) \cdot (x - c)^n}{n!}\).
• For \(\sin(2x)\), the series uses odd powers of \(2x\) due to the alternating pattern of the sine function's derivatives.
• Given the observed pattern, the derived Taylor series is:

\[ f(x) = 2x - \frac{(2x)^3}{3!} + \frac{(2x)^5}{5!} - \frac{(2x)^7}{7!} + \ldots = \sum_{n=0}^{\infty} \frac{(-1)^n \cdot(2x)^{2n+1}}{(2n+1)!} \]This infinite series provides a highly accurate representation of \(\sin(2x)\) near \(x=0\), essential for both mathematical analysis and practical applications.