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A Taylor polynomial is a clever way to approximate complicated functions using polynomials, which are much easier to work with. Polynomials are sums of terms consisting of a variable raised to a power and multiplied by a coefficient.
Let’s consider how a Taylor polynomial is constructed! We choose a point, often denoted as \( a \), and then use the function's value and derivatives at this point. This helps form an approximate polynomial that represents the function near \( a \).
For example, the third-degree Taylor polynomial for the function \( f(x) = \ln(1 + 2x) \) at \( a = 1 \) is obtained by calculating derivative terms at \( a \). In this instance, we use the polynomial:

• Constant term: \( f(1) = \ln(3) \)
• First derivative term: \( f'(1) \times (x-1) = \frac{2}{3}(x-1) \)
• Second derivative term: \( \frac{f''(1)}{2!} \times (x-1)^2 = -\frac{2}{9}(x-1)^2 \)
• Third derivative term: \( \frac{f'''(1)}{3!} \times (x-1)^3 = \frac{8}{81}(x-1)^3 \)

By summing these terms, we construct the polynomial:\[ T_3(x) = \ln(3) + \frac{2}{3}(x-1) - \frac{2}{9}(x-1)^2 + \frac{8}{81}(x-1)^3 \]This polynomial helps in estimating \( f(x) = \ln(1 + 2x) \) around \( x = 1 \).

The remainder of a Taylor polynomial provides insight into the accuracy of the approximation. When using such an approximation, we need to know how much the polynomial diverges from the actual function.
This is where the remainder estimate comes into play. The remainder, \( |R_n(x)| \), is essentially the error between the Taylor polynomial and the actual function for a specific degree \( n \). Calculating it allows us to understand how precise our polynomial approximation is in a given interval.
Using Taylor's inequality, we can estimate the remainder:\[ |R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1} \]Where:

• \( M \) is the maximum value of the next derivative (in this case, the fourth derivative) over the interval.
• \( (n+1)! \) is the factorial of \( n+1 \).

In our scenario, \( f^{(4)}(x) = -\frac{96}{(1+2x)^4} \) reaches a maximum absolute value of approximately 0.64 over \([0.5, 1.5]\). Thus, estimate:\[ |R_3(x)| \leq \frac{0.64}{24} = 0.0267 \]This indicates that the polynomial is quite close within this small error bound.

Calculating derivatives is key to forming a meaningful Taylor polynomial. Derivatives represent how a function changes at any given point, which helps determine the polynomial's coefficients.
For the function \( f(x) = \ln(1 + 2x) \), we find its derivatives as follows:

• First derivative, \( f'(x) = \frac{2}{1 + 2x} \), tells us the slope of the function.
• Second derivative, \( f''(x) = -\frac{4}{(1 + 2x)^2} \), indicates how the slope changes.
• Third derivative, \( f'''(x) = \frac{16}{(1 + 2x)^3} \), provides further details on concavity.
• Fourth derivative, \( f^{(4)}(x) = -\frac{96}{(1 + 2x)^4} \), adds more precision to our calculation.

We calculate derivatives at the point \( a = 1 \) to extract specific values:

• \( f(1) = \ln(3) \)
• \( f'(1) = \frac{2}{3} \)
• \( f''(1) = -\frac{4}{9} \)
• \( f'''(1) = \frac{16}{27} \)

These calculated derivatives are vital components, forming the building blocks of the Taylor polynomial.

Taylor's inequality is a powerful tool that helps us determine how accurate a Taylor polynomial approximation is.
It provides an upper bound for the unexplored error between the function and the polynomial over a specific interval.

• Taylor's inequality formula: \[ |R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1} \]
• \( M \) signifies the maximum value of the derivative's absolute value, here the fourth derivative in the interval \([0.5, 1.5]\).

This inequality serves to assure us about how much trust we can put into our polynomial approximation. It requires evaluating the maximum absolute value \( M \) of the next derivative.
For example, for \( f^{(4)}(x) = -\frac{96}{(1+2x)^4} \), \( M \) is approximately 0.64 within the given interval.
Then, applying:\[ |R_3(x)| \leq \frac{0.64}{24} \approx 0.0267 \]This bound confirms that the Taylor polynomial approximates the function within a small error, aiding us in judging and trusting the polynomial's use for approximations.