Problem 21 - Thomas Calculus

Question

Find the Maclaurin series for the functions \(x^{4}-2 x^{3}-5 x+4\)

Step-by-Step Solution

Answer

The Maclaurin series is \( 4 - 5x - 2x^3 + x^4 \).

Step 1: Understand the Maclaurin Series

A Maclaurin series is a Taylor series expansion of a function about 0, given by: \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \). For polynomial functions, the series becomes finite because higher-order derivatives beyond the degree of the polynomial will be zero.

Step 2: Find the Derivatives

To find the Maclaurin series expansion of \( f(x) = x^{4}-2x^{3}-5x+4 \), we calculate the derivatives at \( x = 0 \). First derivative: \( f'(x) = 4x^3 - 6x^2 - 5 \) Second derivative: \( f''(x) = 12x^2 - 12x \) Third derivative: \( f'''(x) = 24x - 12 \) Fourth derivative: \( f^{(4)}(x) = 24 \) Fifth derivative and beyond will be zero because the polynomial only has degree 4.

Step 3: Evaluate Derivatives at 0

Now evaluate each of these derivatives at \( x=0 \): \( f(0) = 4 \)\( f'(0) = -5 \)\( f''(0) = 0 \)\( f'''(0) = -12 \)\( f^{(4)}(0) = 24 \).

Step 4: Construct the Series

Substitute these values into the Maclaurin series formula: \[ f(x) = 4 + (-5)x + \frac{0}{2!}x^2 + \frac{-12}{3!}x^3 + \frac{24}{4!}x^4 \]Simplify this to:\[ f(x) = 4 - 5x - 2x^3 + x^4 \].

Key Concepts

Taylor Series

A Taylor Series is a powerful mathematical tool that allows us to approximate functions using an infinite sum of terms calculated from the function's derivatives at a single point. In general, a Taylor Series is defined around a point "a" as:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \)

In the specific case of a Maclaurin Series, "a" is zero, thus the expansion becomes:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots \)

This series helps in expressing complex functions in terms of more straightforward polynomial expressions, making them easier to analyze and compute.

Polynomial

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. In general, a polynomial is written as:

• \( a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \)

Polynomials are characterized by their degree, which is the highest power of the variable in the expression. For example, in the polynomial \( x^4 - 2x^3 - 5x + 4 \), the degree is four. These expressions are often straightforward to differentiate and integrate, and are fundamental to algebra and calculus.The reason why a polynomial such as \( x^4 - 2x^3 - 5x + 4 \) results in a finite Maclaurin Series is because the derivatives beyond its highest degree—fourth in this case—are zero. This simplifies calculations as no terms exist beyond those initial derivatives.

Derivative

The derivative of a function is a central concept in calculus that measures how a function changes as its input changes. It provides the rate of change or the slope of the function at a particular point. The derivative of a function \( f(x) \) with respect to \( x \) is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).Calculating derivatives is essential for constructing Taylor or Maclaurin series. It involves successively finding the first, second, third derivatives, and so on. Take, for instance, the polynomial function \( f(x) = x^4 - 2x^3 - 5x + 4 \):

• The first derivative, \( f'(x) = 4x^3 - 6x^2 - 5 \), shows how the polynomial changes per unit of \( x \).
• The second derivative, \( f''(x) = 12x^2 - 12x \), provides information on the concavity of the graph.
• Third derivative, \( f'''(x) = 24x - 12 \), gives insights into how the rate of change itself is changing.
• The fourth derivative, \( f^{(4)}(x) = 24 \), indicates changes at a higher level of calculus insight but for polynomials of degree 4, these are the limits of necessary derivatives.

The derivatives evaluated at \( x = 0 \) are used to build the Taylor and Maclaurin series, highlighting how derivatives are fundamental to these mathematical representations.