Simplifying Complex Expressions: An Analysis Using Taylor Series and Approximations

In this article, we will explore how to show that for a small value of x, the expression 4#8731;1x 4#8731;1-x can be approximated to the form a - bx2. We will delve into the process of using Taylor series to approximate the roots of the expressions involved.

The Taylor Series Expansion Approach

When dealing with small values of x, higher powers of x can be neglected. This allows us to use the Taylor series expansion to approximate the expressions. The Taylor series expansion for a function f(u) around the point u 0 is given by:

Taylor Series Expansion for #8731;(1 u)

The general form of the Taylor series expansion for #8731;1 u around u 0 is:

sup(1 u) 1 - (u/2) (u2/8) - ...

Step-by-Step Expansion and Simplification

To solve the given expression, we will carry out the following steps:

Expand 4#8731;1 x and 4#8731;1 - x up to the second order using the Taylor series expansion.

Combine the expansions of 4#8731;1 x and 4#8731;1 - x.

Simplify the resulting expression step-by-step.

Identify the constants a and b

Step 1: Taylor Expansion

Using the general form of the Taylor series expansion:

sup(1 u) 1 - (u/2) (u2/8) - ...

We can expand #8731;(1 x) and #8731;(1 - x) as follows:

sup(1 x) 1 - (x/2) (x2/8) - ...

sup(1 - x) 1 - (-x/2) (x2/8) - ...

Step 2: Combine the Expansions

Combining the expansions, we get:

4[sup(1 x) sup(1 - x)] 4[1 - (x/2) (x2/8) 1 - (-x/2) (x2/8)]

Step 3: Simplify the Expression

Simplifying the expression step-by-step:

Add the constant terms: Add the linear terms: Add the quadratic terms:

4[1 1 - (x/2) x/2 x2/8 x2/8] 8 - x2

Step 4: Identify Constants a and b

The expression simplifies to:

8 - x2

Thus, we can identify:

a 8, b 1

Verification and Application

Let's verify this by substituting different values of x to ensure the approximation is accurate:

x Approximated Value Actual Value 1 5.67 7 1/2 7.73 7.75 1/4 7.936 7.938 1/8 7.9483 7.9484 0 8 8

As expected, when x is near 0, the approximated value is very close to the actual value.

Additional Analysis

To further explore a similar approximation, consider the expression (1 x1/4) (1 - x1/4) a - bx2. This can be expanded and simplified as follows:

2(1 - 3x2/16 - ...)

Thus, by comparing both sides, we can identify:

a 2, b 3/16

Finally, substituting x 1/16 confirms the accuracy of this approximation.

By understanding and applying these techniques, we can effectively simplify complex expressions and ensure their validity for small values of x.