Problem 44 - Thomas Calculus

Question

The estimate \(\sqrt{1+x}=1+(x / 2)\) is used when \(x\) is small. Estimate the error when \(|x|<0.01\)

Step-by-Step Solution

Answer

The error is at most 0.0000125 when \(|x| < 0.01\).

Step 1: Understanding the Problem

We want to find the error in the estimate \( \sqrt{1+x} = 1 + \frac{x}{2} \) when \(|x| < 0.01\). The error, or the difference between the actual and the estimated value, is given by the absolute value \( |\sqrt{1+x} - (1+\frac{x}{2})| \).

Step 2: Approximate the Actual Value

The actual value is \( \sqrt{1+x} \), so we need to find a way to express this value in a more manageable form. We can iterate an approximation using the Taylor series, but first note that for small \(x\), \(\sqrt{1+x} \approx 1 + \frac{x}{2} - \frac{x^2}{8} + \cdots \).

Step 3: Calculate the Error Term

Since we are using the linear approximation, the error can be approximated by the next term in the Taylor series. In this case, that term is \( -\frac{x^2}{8} \). The error in estimation is primarily this second-order term when \(|x| < 0.01\).

Step 4: Estimate Maximum Error

The maximum value of \(|x|^2\) when \(|x| < 0.01\) is \(0.01^2 = 0.0001\). Thus, the error \( \left| -\frac{x^2}{8} \right| \leq \frac{0.0001}{8} = 0.0000125\).

Step 5: Conclusion

The error in the estimate is at most \(0.0000125\) when \(|x| < 0.01\).

Key Concepts

Taylor Series

The Taylor series is a powerful mathematical tool used for approximating complex functions. It allows us to express a function as an infinite sum of its derivatives at a single point. This comes in handy when the function is too complicated to work with directly. The general form of a Taylor series around the point \( x = a \) is:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots \]For small values of \(x\), as in our exercise, you can often stop at the first few terms for a good approximation.

• The first term gives the value of the function at point \(a\).
• The following terms add information about the function's slope and curvature.

Stopping the series early gives us a simpler approximation, but it introduces an error term. In the exercise, the Taylor series for \( \sqrt{1+x} \) stops after a few terms, primarily using the first-order term to estimate the function.

Linear Approximation

Linear approximation uses the first-order term of a Taylor series to estimate the value of a function around a point. It's a pragmatic approach when you need a fast and easy estimate. For small values of \(x\), the function \(f(x)\) can be approximated by:\[f(x) \approx f(a) + f'(a)(x-a)\]This is essentially the equation of the tangent line of the function at \(x = a\). In our context:

• \(\sqrt{1+x}\) is approximated by the simple linear expression \(1 + \frac{x}{2}\).
• This equation is straightforward, requiring only basic multiplication and addition.

However, keep in mind that linear approximations become less accurate as \(x\) moves away from the central point \(a\). Yet for our purposes, with \(|x| < 0.01\), this estimate is quite reliable.

Second-Order Term

The second-order term in a Taylor series is crucial for understanding the error in linear approximations. This term accounts for the curvature of the function and indicates how the linear approximation deviates from the actual function. In the context of our exercise, the second-order term is:\[-\frac{x^2}{8}\]This term helps us estimate the error in our linear approximation of \(\sqrt{1+x}\). Specifically:

• The error between the linear approximation and the actual function is approximately \(-\frac{x^2}{8}\).
• For very small values of \(x\), this term becomes insignificantly tiny, which supports the effectiveness of the linear approximation.

By calculating this term with a known upper limit for \(x\), we ensure that our error estimation remains accurate and manageable. In this case, with \(|x| < 0.01\), the error is minimal, reaffirming the reliability of our estimation.