Problem 38 - Thomas Calculus

Question

If \(\cos x\) is replaced by \(1-\left(x^{2} / 2\right)\) and \(|x|<0.5,\) what estimate can be made of the error? Does \(1-\left(x^{2} / 2\right)\) tend to be too large, or too small? Give reasons for your answer.

Step-by-Step Solution

Answer

The error is about 0.0026, and the approximation tends to be too small.

Step 1: Understand Taylor Series

The Taylor series expansion for \( \cos x \) around \( x = 0 \) is given by \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \). The expression \( 1 - \frac{x^2}{2} \) is the first two terms of this series. By truncating the series, we introduce an error that consists of all the omitted terms (starting from \( \frac{x^4}{24} \)).

Step 2: Calculate the Next Term for Error Estimate

The next term in the Taylor series after the terms in the approximation \( 1 - \frac{x^2}{2} \) is \( \frac{x^4}{24} \). Since \(|x|<0.5\), the error can be estimated by evaluating this term at the maximum value of \(x\), which is 0.5. The estimated error is \( \left(\frac{0.5^4}{24}\right) = \frac{0.0625}{24} \approx 0.0026 \).

Step 3: Analyze Error Direction

Because the error term \( \frac{x^4}{24} \) is positive, the approximation \( 1 - \frac{x^2}{2} \) will always be less than \( \cos x \) for \(|x|<0.5\). This implies that the approximation tends to be too small, as you are removing this positive quantity from the true answer.

Key Concepts

Cosine Approximation

To approximate the cosine function, we often use the Taylor series expansion. For small values of \(x\), the cosine function can be expressed as an infinite series:

• \( \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720} + \cdots \)

When we seek a simple approximation, we truncate the series to only a few terms. In this case, the expression \(1 - \frac{x^2}{2}\) is used.
This truncated series captures the cosine function's behavior for small \(x\) but will not be perfect. The truncated version gives us a balance between accuracy and simplicity, ideal for situations where a precise evaluation of several terms is too complex or time-consuming.
In practice, this means that the cosine function can be quickly and easily approximated for small angles by just using this simpler expression.

Error Estimation

When approximating functions by truncating a Taylor series, estimating the error is crucial. The error tells us how far our approximation is from the actual value of the function.
For the cosine function, the first term left out when we truncate at \(1 - \frac{x^2}{2}\) is \(\frac{x^4}{24}\). Evaluating this term provides an estimate for the error incurred by not including subsequent terms in the series.

• For \(|x| < 0.5\), the error term becomes \(\left(\frac{0.5^4}{24}\right) = \frac{0.0625}{24} \approx 0.0026\).

This calculation shows that for inputs within this range, the approximation error is about 0.0026. This suggests that while the approximation is not perfect, it remains fairly close to the actual cosine value, even with just two terms from the series.

Truncation Error

Truncation error occurs when we stop the series at a certain point, which means ignoring all later terms. This error helps us understand whether our approximation is over or underestimating the true value.
Analyzing the truncation error direction is essential for recognizing if the approximation tends to be too high or too low. In this example:

• The term \(\frac{x^4}{24}\) is positive, indicating that our truncated series expression \(1 - \frac{x^2}{2}\) is always less than the actual \(\cos x\).

Thus, for \(|x| < 0.5\), our approximation consistently underestimates the true cosine value. This tells us that whenever you use \(1 - \frac{x^2}{2}\) to approximate \(\cos x\) in this range, the result will be slightly smaller than the actual cosine function value.