Determine the truncation in given that $$ \sqrt{0.98} $$ = (1 - 0.02)$$ ^{0.5} $$

To determine the truncation error when approximating $ \sqrt{0.98} $ using the expression $ (1 - 0.02)^{0.5} $, let's start by employing the binomial series expansion and then compare it with the exact value.

### Binomial Series Expansion

For small $x$, the binomial series expansion for $(1 - x)^n$ where $n = 0.5$ is given by:

\[
(1 - x)^{0.5} = 1 - \frac{x}{2} - \frac{x^2}{8} + \frac{5x^3}{48} - \cdots
\]

Here, $x = 0.02$ and we want to approximate $(1 - 0.02)^{0.5}$. Let's expand this using the first few terms of the binomial series:

\[
(1 - 0.02)^{0.5} \approx 1 - \frac{0.02}{2} - \frac{(0.02)^2}{8} + \frac{5(0.02)^3}{48}
\]

Calculating each term separately:
\[
\frac{0.02}{2} = 0.01
\]
\[
\frac{(0.02)^2}{8} = \frac{0.0004}{8} = 0.00005
\]
\[
\frac{5(0.02)^3}{48} = \frac{5(0.000008)}{48} = \frac{0.00004}{48} \approx 0.000000833
\]

Adding these terms up to the cubic term:
\[
(1 - 0.02)^{0.5} \approx 1 - 0.01 - 0.00005 + 0.000000833 \approx 0.989950833
\]

### Exact Value Calculation

The exact value of $ \sqrt{0.98} $ can be calculated as follows:

\[
\sqrt{0.98} \approx 0.9899494937
\]

### Truncation Error

The truncation error when truncating after the linear term can be calculated as:

Using only the first term (linear approximation):
\[
(1 - 0.02)^{0.5} \approx 1 - 0.01 = 0.99
\]

- **Truncated value (linear term only)**: 0.99
- **Exact value**: $ \sqrt{0.98} \approx 0.9899494937 $

The truncation error is the difference between the exact value and the truncated series approximation (using only the linear term):

\[
\text{Truncation Error} = \sqrt{0.98} - (1 - 0.01) = 0.9899494937 - 0.99 = -0.0000505063
\]

### Relative Error

The relative error can be calculated as:

\[
\text{Relative Error} = \frac{\text{Truncation Error}}{\text{Exact Value}} = \frac{-0.0000505063}{0.9899494937} \approx -0.000051
\]

### Summary

- **Truncated value (linear term only)**: $ (1 - 0.02)^{0.5} \approx 0.99 $
- **Exact value**: $ \sqrt{0.98} \approx 0.9899494937 $
- **Truncation Error**: $ -0.0000505063 $
- **Relative Error**: $ -0.000051 $

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What is a binomial series expansion and when is it useful?
2. How does the binomial expansion apply to fractional exponents?
3. Why do higher-order terms in the expansion improve accuracy?
4. What are the applications of truncation error analysis in numerical methods?
5. How can we determine the optimal number of terms to include in a series expansion for a given accuracy?
6. What is the difference between truncation error and round-off error?
7. How does the choice of $x$ affect the accuracy of binomial expansions?
8. What are alternative methods for approximating square roots and their respective errors?

**Tip:** When using series expansions for approximations, consider including higher-order terms for better accuracy, especially when higher precision is required.

Learn how to calculate the truncation error when approximating √0.98 using a binomial series expansion. This problem covers key concepts in numerical methods and approximation techniques. Suitable for advanced high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation