Question

Use Maclaurin series to evaluate: $1-{\sqrt{1-x^2}|}_{x={10}^{-6}}$

Short answer

$5\times {10}^{-13}$

Step-by-step solution

- Recall the Maclaurin series for the square root function

The Maclaurin series for $\text{Could not find of for root}$ can be written as $$\text{Could not find of for root}$$

- Substitute $x={10}^{-6}$ into the series

Now that the series is known, substitute x=10^{-6} into the series to evaluate the approximation: $$\sqrt{1-x^2}\approx 1-\frac{({10}^{-6}{)}^2}{2}-\frac{({10}^{-6}{)}^4}{8}$$

- Simplify the approximation

Evaluate the terms: $$1-\frac{({10}^{-6}{)}^2}{2}-\frac{({10}^{-6}{)}^4}{8}=1-\frac{{10}^{-12}}{2}-\frac{{10}^{-24}}{8}\approx 1-5\times {10}^{-13}$$

- Find the final expression

Now, compute the value of \1 - ( \sqrt{1 - x^2})\: $$1-(1-5\times {10}^{-13})=5\times {10}^{-13}$$

Key concepts

Series Expansion

A series expansion is a way to express a mathematical function as a sum of terms. Each term in the series typically involves a power of a variable. Maclaurin series are a specific type of series expansion that represents a function as a power series centered at zero.

For instance, the Maclaurin series for the square root function $\sqrt{1-x}$ can be written as: $$\sqrt{1-x}\approx 1-\frac{x}{2}-\frac{x^2}{8}-...$$. This is very useful when the value of x is small, as higher-order terms become negligible.

In the given exercise, we used this concept to approximate $\sqrt{1-x^2}$ by considering only the first few terms of the series.

• The series provides a straightforward approximation method when exact calculations are complex.
• It helps simplify complicated functions into manageable forms.

Approximation Methods

Approximation methods are techniques used to find approximate values of complex mathematical expressions. These methods are especially useful when dealing with functions that are difficult to compute exactly.

Maclaurin series is a common approximation technique. By expanding a function into a series, we can approximate its value at specific points.

One benefit of using approximation methods like the Maclaurin series is that they offer a controlled way to achieve the desired level of accuracy. For instance, in our exercise, substituting x=10^-6 into the series:

• $$\sqrt{1-x^2}\approx 1-\frac{({10}^{-6}{)}^2}{2}-\frac{({10}^{-6}{)}^4}{8}=1-5\times {10}^{-13}$$
• The error margin can be controlled by considering more terms in the series.

Small Angle Approximation

The small angle approximation is a technique where functions involving small values of x are simplified. It is often used in trigonometry and other branches of mathematics.

When x is very small, higher powers of x become negligible. For example, $x^2$, $x^4$ are much smaller than x itself. In our exercise, given that x = 10^-6, higher-order terms like $x^4$ can be disregarded.

This makes complex expressions much simpler. For example:

Using the small angle approximation for our series:

• $$\sqrt{1-x^2}\approx 1-\frac{x^2}{2}-\frac{x^4}{8}\approx 1-\frac{x^2}{2}$$
• This simplified our calculations, allowing us to find: $5\times {10}^{-13}$

Small angle approximations are crucial for making otherwise complicated calculations manageable.

Mathematical Analysis

Mathematical analysis is a branch of mathematics focused on limits and the behavior of functions. It provides the foundation for calculus and other advanced topics. Using concepts from mathematical analysis, we can evaluate how functions behave under various conditions.

In the exercise, we relied on principles of mathematical analysis to use the Maclaurin series for approximating $\sqrt{1-x^2}$ when x is a small value $({10}^{-6})$.

• This approach allows us to understand the behavior of the function at specific and often very small intervals.
• It simplifies complicated expressions and makes them calculable.

Analyzing functions in this way is crucial for developing efficient and precise mathematical models.