Problem 18 - Calculus: Early Transcendentals

Question

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = \ln(1 + 2x), \) \( a = 1, \) \( n = 3, \) \( 0.5 \le x \le 1.5 \)

Step-by-Step Solution

Answer

The Taylor polynomial is approximately \( \ln(3) + \frac{2}{3}(x-1) - \frac{2}{9}(x-1)^2 + \frac{8}{81}(x-1)^3 \). The error is around 0.024.

Step 1: Determine the Taylor Polynomial

We need to find the Taylor polynomial of degree 3 for \( f(x) = \ln(1+2x) \) centered at \( a = 1 \). First, compute the derivatives of \( f(x) \). - The 0th derivative: \( f(x) = \ln(1+2x) \).- The 1st derivative: \( f'(x) = \frac{2}{1+2x} \).- The 2nd derivative: \( f''(x) = \frac{-4}{(1+2x)^2} \).- The 3rd derivative: \( f'''(x) = \frac{16}{(1+2x)^3} \).Evaluate each derivative at \( x = 1 \):- \( f(1) = \ln(3) \)- \( f'(1) = \frac{2}{3} \)- \( f''(1) = \frac{-4}{9} \)- \( f'''(1) = \frac{16}{27} \)The Taylor polynomial is given by:\[ T_3(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3 \]Compute each term:\( T_3(x) = \ln(3) + \frac{2}{3}(x-1) + \frac{-2}{9}(x-1)^2 + \frac{8}{81}(x-1)^3 \).

Step 2: Use Taylor's Inequality

To estimate the error of approximation, use Taylor's Inequality:\[ | R_3(x) | \leq \frac {M |x-a|^{4}} {4!} \]where \( M \) is an upper bound on the absolute value of the 4th derivative on the interval between \( a = 1 \) and any \( x \) in \( [0.5, 1.5] \).Find the 4th derivative:- \( f^{(4)}(x) = \frac{-96}{(1+2x)^4} \)Estimate the maximum value of \( |f^{(4)}(x)| \) for \( 0.5 \leq x \leq 1.5 \):- The largest value occurs at \( x = 0.5 \), giving \( M = \frac{192}{625} \).Thus,\( | R_3(x) | \leq \frac{192}{625} (0.5)^{4} \), approximately 0.024.

Step 3: Graph \( |R_3(x)| \)

To verify the result from Step 2, plot the error term \( |R_3(x)| = |f(x) - T_3(x)| \) over the interval \( 0.5 \le x \le 1.5 \). Use a graphing tool or software.Check that the error values on the graph are consistent with the maximum error bound obtained using Taylor's Inequality in Step 2. The graph should show that the error does not exceed approximately 0.024 throughout the interval.

Key Concepts

Taylor's Inequality

Taylor's Inequality is a useful tool for estimating the error when using a Taylor polynomial to approximate a function. It gives an upper bound on the error so that you can know how close your approximation is to the real function.
To apply Taylor's Inequality, you usually look at the next derivative beyond the degree of your polynomial. For instance, when using a third-degree polynomial, you need the fourth derivative.
The inequality is expressed as:

• \[ | R_n(x) | \leq \frac {M |x-a|^{n+1}} {(n+1)!} \]

Here, \( R_n(x) \) is the approximation error, \( M \) is the maximum value of the \((n+1)\)-th derivative on the interval, and \( a \) is the center of the Taylor polynomial.
This formula shows that the error decreases as \(|x-a|\) becomes smaller.

Approximation Accuracy

Approximation accuracy describes how closely a Taylor polynomial matches the actual function it is approximating. Higher degree polynomials can provide better approximations because they consider more terms of the function's Maclaurin or Taylor series.
The accuracy is often evaluated within a specific interval around the center \(a\). The closer the value of \(x\) to \(a\), the more accurate the Taylor polynomial will generally be.
Using Taylor's Inequality, you can calculate the maximum error and understand the worst-case discrepancy between the polynomial approximation and the actual function. This is particularly useful in ensuring that your approximation is reliable for your planned use.
Visualizing the approximation and the actual function on a graph can also help you see where the polynomial matches well and where it diverges.

Derivatives

Derivatives play a critical role in constructing Taylor polynomials. They determine the coefficients of the polynomial, and thus directly influence both the shape of the polynomial and its approximation accuracy.
When calculating a Taylor polynomial, you first need to find several derivatives of the function. For a third-degree polynomial, compute up to the third derivative. Insert these into the Taylor series formula to find polynomial-specific terms.
For example, if the function is \( f(x) = \ln(1+2x)\) and you want a polynomial expansion at \(a = 1\), you first find derivatives like \( f'(x) \), \( f''(x) \), and \( f'''(x) \). Evaluating these derivatives at \(x = 1\) gives the coefficients for the polynomial, which provide insights into how the function behaves in the local region.

Error Estimation

Whenever you use a Taylor polynomial to approximate a function, you need an estimate of how far off your approximation could be. Error estimation helps you identify this difference, ensuring the approximation is within acceptable limits.
Using Taylor's inequality allows for this estimation as it calculates an upper bound error. For instance, the error of a third-degree polynomial might be approximated by factoring in the fourth derivative of the function.

• The 4th derivative helps identify the worst-case scenario of the polynomial's deviation from the actual function over a given interval.
• By evaluating this derivative throughout the interval, you find the maximum possible error, ensuring the approximation stays within this anticipated range.

Knowing this helps verify if the approximation meets required precision, making it easier to apply this in practice for calculations or predictions.