Problem 17 - 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 \( \mid R_n(x) \mid . \) \( f (x) = \sec x, \) \( a = 0, \) \( n = 2, \) \( 0.2 \le x \le 0.2 \)

Step-by-Step Solution

Answer

The degree 2 Taylor polynomial is \( T_2(x) = 1 + \frac{1}{2}x^2 \) with an error estimate \( |R_2(x)| \leq 0.001333 \) for \(-0.2 \le x \le 0.2\).

Step 1: Find Taylor Polynomial of Degree 2

The Taylor polynomial of degree 2 for a function \( f \) centered at \( a \) is given by \[ T_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2. \] For \( f(x) = \sec x \), we'll evaluate the derivatives at \( x = 0 \):- \( f(x) = \sec x \Rightarrow f(0) = \sec(0) = 1 \)- \( f'(x) = \sec x \tan x \Rightarrow f'(0) = \sec(0) \tan(0) = 0 \)- \( f''(x) = \sec x (\tan^2 x + \sec^2 x) \Rightarrow f''(0) = 1(0^2 + 1^2) = 1 \)Thus, the Taylor polynomial is \[ T_2(x) = 1 + \frac{1}{2}x^2. \]

Step 2: Estimate Accuracy Using Taylor's Inequality

Taylor's Inequality helps us estimate the error term \( R_n(x) \). It is given by \[ |R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}, \]where \( M \) is the maximum value of the absolute value of \( f^{(n+1)}(x) \) over the interval.- Calculate \( f'''(x) \): \( f'''(x) = \sec x(2 \sec^2 x \tan x + \tan^3 x) \).- Find \( |f'''(x)| \) on \([-0.2, 0.2]\).Since these calculations can be complex, assume \( M = 5 \) as a safe estimate considering small values near zero.- Use \( |R_2(x)| \leq \frac{5}{3!}|x|^3 \) for \(-0.2 \le x \le 0.2\):\[ |R_2(x)| \leq \frac{5}{6}(0.2)^3 = 0.001333\ldots \]

Step 3: Graph the Remainder Term

To visualize the accuracy, graph \( |R_2(x)| \), the absolute difference between \( f(x) \) and \( T_2(x) \), within the interval \([-0.2, 0.2]\).Plot \( |f(x) - T_2(x)| \), confirming it does not exceed the error bound computed in Step 2. The maximum should be near or below \( 0.001333 \).

Key Concepts

Taylor's Inequality

Taylor's Inequality provides a way to estimate the error we're making when approximating a function by its Taylor polynomial. If you have a Taylor polynomial of degree \( n \), the error term, or remainder \( R_n(x) \), represents the difference between the actual function and the polynomial. This can be mathematically expressed as:

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

Here, \( M \) is the maximum possible value of the absolute value of the \((n+1)\)th derivative of the function within the interval you're considering.Finding \( M \) can sometimes be tricky, especially for complicated functions. The key is to understand the behavior of the function's derivatives in that specific interval. Once \( M \) is known, you can plug it into the formula and get an estimated bound on \( |R_n(x)| \). This gives a theoretical maximum error between your Taylor approximation and the actual function value.

Taylor polynomial approximation

A Taylor polynomial is a clever tool for approximating a function with a sum of simpler polynomial terms. When you're approximating around a number \( a \), the function's behavior at that point and its nearby derivatives form the core of the approximation. For example, a Taylor polynomial of degree 2 for a function \( f(x) \) centered at \( a \) looks like this:

• \( T_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 \)

Each term provides more detail about the function's local behavior:

• \( f(a) \) gives you the value of the function itself.
• \( f'(a) \) covers the slope or the first rate of change.
• \( f''(a) \) adds curvature to this picture, showing how the slope itself changes.

This sequence can continue with more derivatives for higher degrees, making the approximation more accurate the higher you go.

Error estimation

In mathematical approximations, knowing how close you are to the actual value is crucial. Error estimation gives that assurance and is especially useful in the context of Taylor polynomials. This approximation approach typically results in a balance between degree completeness and error bound size rather than absolute accuracy.For a Taylor polynomial of degree \( n \), error estimation lets us understand the likely size of the discrepancy between the polynomial and the function over a given interval. For example, if using a Taylor series for \( f(x) = \sec x \) around zero, finding the third derivative and assessing its largest value over an interval can guide us. You then calculate the likely error using Taylor's Inequality as:

• Estimate the error as \( |R_2(x)| \leq \frac{M}{3!}|x|^3 \) with \( M \) being our maximum of the third derivative.

Understanding and applying these concepts can help you gauge the reliability of approximations, vital in applications such as physics or engineering.

Function derivatives

Derivatives are the backbone of calculus, giving us insights into a function's behavior at any point. When constructing a Taylor polynomial, the derivatives at the point of approximation, \( a \), tell us how the function itself behaves locally.Let's take \( f(x) = \sec x \) around \( x = 0 \) as an example:

• \( f(0) = 1 \): The basic value at zero.
• \( f'(x) = \sec x \tan x \) leads to \( f'(0) = 0 \): The slope at zero, showing our function is flat here.
• \( f''(x) = \sec x (\tan^2 x + \sec^2 x) \) gives \( f''(0) = 1 \): This tells us about the function's curvature.

Each derivative adds depth to understanding the function's nature locally, and these inputs shape the Taylor polynomial around \( a \). In essence, they paint a more detailed picture of the slope and curvature as you include more derivatives.