Problem 11 - Calculus

Question

Find the first four terms of the Taylor series for the function about the point \(a\). $$\sin \theta, \quad a=-\pi / 4$$

Step-by-Step Solution

Answer

The series is \(-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\theta + \pi/4) + \frac{\sqrt{2}}{4}(\theta + \pi/4)^2 - \frac{\sqrt{2}}{12}(\theta + \pi/4)^3\).

Step 1: Identify the Taylor Series Formula

A Taylor series for a function \(f(x)\) about a point \(a\) is given by:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots\]We need to apply this formula to the function \(\sin \theta\) around \(a = -\pi/4\).

Step 2: Evaluate the Function and the Derivatives at \(a\)

First, we calculate the value of the function and its derivatives at \(a = - \pi / 4\):- \(f(\theta) = \sin \theta\), so \(f(-\pi/4) = \sin(-\pi/4) = -\frac{\sqrt{2}}{2}\).- The first derivative is \(f'(\theta) = \cos \theta\); thus, \(f'(-\pi/4) = \cos(-\pi/4) = \frac{\sqrt{2}}{2}\).- The second derivative is \(f''(\theta) = -\sin \theta\); thus, \(f''(-\pi/4) = -\sin(-\pi/4) = \frac{\sqrt{2}}{2}\).- The third derivative is \(f'''(\theta) = -\cos \theta\); thus, \(f'''(-\pi/4) = -\cos(-\pi/4) = -\frac{\sqrt{2}}{2}\).

Step 3: Substitute Values into the Taylor Series

Substituting into the Taylor series formula:\[\sin(\theta) \approx -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\theta + \pi/4) + \frac{\sqrt{2}/2}{2!}(\theta + \pi/4)^2 - \frac{\sqrt{2}/2}{3!}(\theta + \pi/4)^3\]Simplifying each term, we get:- First term: \(-\frac{\sqrt{2}}{2}\)- Second term: \(+ \frac{\sqrt{2}}{2}(\theta + \pi/4)\)- Third term: \(+ \frac{\sqrt{2}}{4}(\theta + \pi/4)^2\)- Fourth term: \(- \frac{\sqrt{2}}{12}(\theta + \pi/4)^3\)

Step 4: Write the First Four Terms of the Series

The first four terms of the Taylor series for \(\sin\theta\) around \(a = -\pi/4\) are:\[-\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}(\theta + \pi/4) + \frac{\sqrt{2}}{4}(\theta + \pi/4)^2 - \frac{\sqrt{2}}{12}(\theta + \pi/4)^3\]

Key Concepts

Understanding Derivatives

Derivatives are essential in calculus and mathematical analysis. They represent the rate at which a function is changing at any point, which can be thought of as the slope of the tangent line to the graph of the function at that point.
When we talk about derivatives, these are some key points to remember:

• The first derivative of a function provides the slope of the tangent line and indicates how the function is increasing or decreasing at a given point.
• The second derivative provides information about the concavity of the function and can be used to determine inflection points, where the concavity changes.
• Higher-order derivatives, like the third derivative, continue this pattern of providing deeper insights into the behavior of the function.

In the context of Taylor series, derivatives at a specific point are used to construct an approximation of the function. In the original exercise, derivatives of the sine function up to the third order were needed for accurate approximation.

Function Approximation Using Taylor Series

Taylor series provide a powerful tool for function approximation. They allow a complicated function to be expressed as an infinite sum of simpler polynomial terms.
To construct a Taylor series, we start at a specific point and generate the series using derivatives:

• The function value and its derivatives at a particular point are used to estimate the function over an interval.
• The Taylor series formula combines these values into a polynomial that can approximate the function around this specific point.
• This approximation can be quite accurate near the chosen point and becomes less precise further away from it.

For instance, in the given exercise, Taylor series were employed to approximate the sine function around the point \(a = -\frac{\pi}{4}\). By calculating the function and its derivatives at \(a\), these derivatives formed the coefficients of the polynomial, step-by-step providing an accurate local representation of the sine function.

Exploration of Trigonometric Functions

Trigonometric functions, like sine and cosine, are foundational in many areas of mathematics and science. They describe the relationships between the angles and sides of triangles and are periodic functions that repeat values in regular intervals.
Here are some basic concepts about trigonometric functions:

• The sine function \( \sin \theta \) is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
• Cosine, another essential trigonometric function, represents the ratio of the length of the adjacent side to the hypotenuse.
• These functions are periodic, meaning they repeat their values in regular intervals, a property exploited when using Taylor series to approximate them.

For approximating trigonometric functions using Taylor series, understanding these basics is crucial, as it helps simplify and represent complex wave-like functions. In the original exercise, the sine function's Taylor series was expanded to approximate its behavior near \( -\pi/4 \), demonstrating the beauty and utility of these series.