Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Arrive at the series for \(\cos x\) by differentiating the Maclaurin series for \(\sin x\).

Step-by-Step Solution

Answer

The Maclaurin series for \(\cos x\) is indeed gotten by differentiating the Maclaurin series for \(\sin x\) term by term. The series we obtain from doing this is \[1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots\].

Step 1: Write the Maclaurin Series for \(\sin x\)

The Maclaurin series for \(\sin x\) is given by \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots\). This is an infinite alternate series with \(x^n/n!\) as term.

Step 2: Differentiate the Maclaurin Series for \(\sin x\)

Differentiation term by term gives the series of \(\cos x\). Following the power rule for differentiation (the derivative of \(x^n\) is \(n*x^{n-1}\)), we obtain the following series: \[1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots\]

Step 3: Identify the Derived Series with \(\cos x\)

This series obtained by term-by-term differentiation of \(\sin x\)'s Maclaurin series is the Maclaurin series of \(\cos x\). Hence we showed that differentiating the Maclaurin series for \(sin x\) gives the Maclaurin series for \(\cos x\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, centered around determining the rate at which a function changes at any given point. To differentiate a function means to find its derivative, which provides a new function describing this rate of change. For polynomial functions, the power rule is a straightforward tool: if you have a term like \(x^n\), its derivative is \(n \cdot x^{n-1}\). This rule is exceptionally useful when working with power series.

When applying differentiation to a series, such as the Maclaurin series for \(\sin x\), each term is individually differentiated. In this context, the series is \[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \]

• Apply the derivative term by term.
• Adjust the powers and coefficients according to the power rule.

The result leads to a new series, which in this case gives us the series for \(\cos x\).

Trigonometric Functions

Trigonometric functions such as \(\sin x\) and \(\cos x\) are periodic functions commonly used in various fields like physics and engineering. They relate the angles of a triangle to the lengths of its sides in a right-angled triangle. More generally, they are fundamental in describing wave patterns, oscillations, and other phenomena that repeat cyclically.

The function \(\sin x\) corresponds to the vertical coordinate of a point on the unit circle as the angle \(x\) sweeps around the circle. In contrast, \(\cos x\) gives the horizontal coordinate. They are related through their derivatives:

• The derivative of \(\sin x\) is \(\cos x\).
• The derivative of \(\cos x\) is \(-\sin x\).

This relationship helps in transforming the series of one function into that of another, as evidenced by differentiating the Maclaurin series for \(\sin x\) to get the series for \(\cos x\). It highlights the elegant symmetry between these two important trigonometric functions.

Power Series

A power series is an infinite series of the form \(a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots\), where the coefficients \(a_n\) are constants. Power series are powerful tools in mathematical analysis, allowing complex functions to be expressed as infinite sums of polynomials. They are especially useful for approximating functions and solving differential equations.

The Maclaurin series is a specific type of power series centered at zero. For trigonometric functions like \(\sin x\) and \(\cos x\), the Maclaurin series provide a polynomial approximation of the function:

• For \(\sin x\): \(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots\)
• For \(\cos x\): \(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots\)

These series demonstrate the behavior of trigonometric functions as sums of their polynomial components, allowing us to easily compute function values for small angles \(x\). Understanding power series is essential for grasping the convergence and approximation of functions, which underpins much of calculus and analysis.