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Taylor's Inequality is a crucial tool used to estimate the error or remainder when approximating a function with a Taylor polynomial. When you approximate a function, like \( f(x) = \sec x \), by a Taylor polynomial of degree \( n = 2 \), Taylor's Inequality helps you understand how close your approximation is to the actual function across a specific interval. In this case, the interval is \([-0.2, 0.2]\).
The formula given by Taylor's Inequality for the remainder \( R_n(x) \) is:

• \( R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} \)

Here, \( f^{(n+1)}(c) \) represents the \((n+1)\)-th derivative of the function evaluated at some value \( c \) in the interval. Estimating this derivative allows you to establish a bound for the approximation error.
For this particular problem, the third derivative \( f^{(3)} \) was determined to have a maximum absolute value of approximately 4 on the interval, leading to the calculation of \(|R_2(x)| \leq \frac{4}{6} (0.2)^3 \approx 0.00267\).

Derivatives play an essential role in constructing Taylor polynomials since they determine the coefficients of the polynomial. For the function \( f(x) = \sec x \), we start by finding the first few derivatives because those are necessary for creating a Taylor polynomial up to degree 2. Let's break down these derivatives and their computation:

• \( f(x) = \sec x \), the basic function.
• \( f'(x) = \sec x \tan x \), the first derivative found using the product rule.
• \( f''(x) = \sec x (\tan^2 x + \sec^2 x) \), the second derivative that combines trigonometric identities.

Evaluating these derivatives at the point \( a = 0 \) gives us:

• \( f(0) = 1 \), since \( \sec 0 = 1 \).
• \( f'(0) = 0 \), since \( \tan 0 = 0 \).
• \( f''(0) = 1 \), as derived by substituting into the second derivative formula.

These values help form the Taylor polynomial \( T_2(x) = 1 + \frac{1}{2}x^2 \).

Remainder estimation involves determining how good an approximation your Taylor polynomial is. In step-by-step calculations, we use Taylor's Inequality to establish a bound on the remainder of our approximation. This gives insights into how close the Taylor polynomial is to the actual function across a specific interval.
The task of finding the remainder \( R_2(x) \) meant calculating the third derivative since we need an \( n+1 \)-th term for a degree \( n = 2 \) approximation. The formula becomes:

• \( R_2(x) = \frac{f^{(3)}(c)}{3!}x^3 \)

The challenge then is to find or estimate \( |f^{(3)}(x)| \) across the interval \([-0.2, 0.2]\). Ultimately, estimating it to be at most 4 allowed computation of the bound \(|R_2(x)| \leq \frac{4}{6} (0.2)^3 \approx 0.00267\), signifying the approximation's accuracy.
This estimation is crucial for determining how reliable our Taylor polynomial is as a representation of \( f(x) = \sec x \).

Graph verification is a practical way to substantiate the theoretical bounds you derive using Taylor's Inequality. By graphing \( |R_2(x)| \) over the interval \([-0.2, 0.2]\), you check visually how the remainder measures against the calculated bound.
To do this, consider the expression \(| f(x) - T_2(x) |\). This will show exactly how much the Taylor polynomial \( T_2(x) = 1 + \frac{1}{2}x^2 \) deviates from the actual function \( f(x) = \sec x \). On the graph, if the top of the plot never exceeds the bound \(0.00267\), the approximation is within acceptable error limits.
Graphing \( |R_2(x)| \) effectively serves as a visual confirmation of the accuracy of the Taylor approximation and provides an immediate, intuitive grasp of the polynomial's performance across the prescribed interval.