To determine the truncation error when approximating $\sqrt{0.98}$ using the expression $(1 - 0.02)^{0.5}$, we can use a series expansion.

Series Expansion

We start by using the binomial expansion for $(1 - x)^{0.5}$. For small $x$, the binomial series expansion of $(1 - x)^n$ where $n = 0.5$ is:

$(1 - x)^{0.5} = 1 - \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \cdots$

For our specific case where $x = 0.02$:

$(1 - 0.02)^{0.5} = 1 - \frac{0.02}{2} - \frac{(0.02)^2}{8} + \frac{(0.02)^3}{16} - \cdots$

Truncating after the linear term:

$(1 - 0.02)^{0.5} \approx 1 - \frac{0.02}{2} = 1 - 0.01 = 0.99$

Exact Value Calculation

Next, let's calculate the exact value of $\sqrt{0.98}$:

$\sqrt{0.98} \approx 0.989949$

Truncation Error

The truncation error is the difference between the exact value and the truncated series approximation:

$\text{Truncation Error} = \sqrt{0.98} - (1 - 0.01) = 0.989949 - 0.99 = -0.000051$

Relative Error

The relative error can be calculated as:

$\text{Relative Error} = \frac{\text{Truncation Error}}{\text{Exact Value}} = \frac{-0.000051}{0.989949} \approx -0.0000515$

The negative sign indicates that the truncated value is slightly higher than the actual value, but the magnitude of the relative error is what is usually of interest.

Summary

• Truncated value: $(1 - 0.02)^{0.5} \approx 0.99$
• Exact value: $\sqrt{0.98} \approx 0.989949$
• Truncation Error: $-0.000051$
• Relative Error: $-0.0000515$

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

1. What is a series expansion and how is it used in approximations?
2. Why is the binomial expansion useful for small values of $x$?
3. How does truncation affect the accuracy of an approximation?
4. What are the differences between absolute error and relative error?
5. How can we improve the accuracy of the binomial approximation?
6. What are other common methods for approximating square roots?
7. How do different orders of truncation affect the error?
8. What is the importance of error analysis in numerical methods?

Tip: Always consider higher-order terms in series expansions to improve the accuracy of approximations.