Question

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

Short answer

Answer: In general, the accuracy of a Taylor polynomial approximation tends to increase with the order of the approximation, as higher-order polynomials can better represent the function around the expansion point. However, the accuracy improvement is highly dependent on the function's higher-order derivatives and its behavior within a specific interval.

Step-by-step solution

Understanding Taylor Polynomials

Taylor polynomials are a way to approximate a function by means of a polynomial that matches its value and derivatives at a single point. The order of the Taylor polynomial refers to the highest power of the variable present in the polynomial. Higher-order polynomials include more terms and hence can capture more complex behavior of the function we are approximating.

The Taylor Series for a Function

The Taylor series of a function f(x) about a point "a" is given by: \[f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots\] The expression above includes an infinite series. However, the Taylor polynomial truncates the series at a certain order. For example, the 2nd-degree Taylor polynomial will only include terms up to (x-a)^2.

Increasing Order of the Approximation

As we add more terms by increasing the order of the Taylor polynomial, we are better able to represent the behavior of the function at the expansion point. In most cases, this results in increased accuracy of the approximation within a certain interval around the expansion point. However, it's essential to notice that the accuracy of the approximation will heavily depend on the radius of convergence and the behavior of the function itself.

The Remainder Term and Error Estimation

Taylor's theorem also includes a remainder term called the Lagrange Remainder, which helps to estimate the error in the approximation. The remainder term is given by: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x - a)^{n+1}\] Here, "n" is the degree of the Taylor polynomial, and "c" is a point between "a" and "x". The remainder term is also affected by the function's higher-order derivatives and the interval between "a" and "x". The error in approximation will decrease as we increase the order of the Taylor polynomial if the higher-order derivatives are converging or if the values of (x-a)^{n+1} are decreasing for larger "n" values.

Conclusion

In general, the accuracy of a Taylor polynomial approximation tends to increase with the order of the approximation. Including more terms in the approximation will allow us to represent the function more accurately around the expansion point. However, the accuracy improvement is heavily reliant on the function's higher-order derivatives and the behavior of the function itself within a specific interval.

Key concepts

Taylor Series

When you're delving into calculus, one of the most brilliant tools at your disposal is the Taylor series. This mathematical marvel allows you to approximate complex functions with polynomials, making difficult problems more manageable. Imagine trying to get a close-up view of a curve in a function. The closer you look, the straighter it seems. That’s the core concept behind the Taylor series - it fits a smooth curve, like a polynomial, to a function’s jagged landscape at a specific point. The basic formula is a symphony of function values and derivatives that sounds like this:
\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \dots \]
This series can be infinitely long, but usually we cut it off after a few terms to create a Taylor polynomial which is easier to work with. If the series converges, we can actually represent some functions exactly! Think of it as a mathematical tailor, shaping a polynomial suit that fits your function just right at the 'a' point.

Order of a Taylor Polynomial

You might be wondering how you know if your Taylor polynomial is giving you a good approximation. That's where the 'order' comes into play. It's like deciding how many pieces of a puzzle you need to see the picture. Each piece in our puzzle is a term in the polynomial.

The order is essentially the degree of the highest power in our polynomial. For example, a 3rd-order Taylor polynomial will venture up to the term with \(x-a)^3\). If you go for more terms, say 5th order, your approximation is like adding extra pieces to your puzzle, usually getting you a clearer image of the original function's curve.

Tip for Studying:

• Start with low-order polynomials to grasp the basic shape of your function.
• Gradually increase the order to see how the approximation improves.
• Be aware that at some point, adding more terms won't significantly improve the approximation due to the limits of convergence.

Lagrange Remainder

Imagine you've ordered a delicious pizza but got a slightly smaller size than what you wanted. The Lagrange Remainder is kind of like that - it's the difference between your Taylor polynomial 'slice' and the whole 'pizza' of your actual function value. You use it to get a handle on the error of your approximation.

It goes like this:
\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x - a)^{n+1} \]
Here, the 'n+1'th derivative of your function helps determine the maximum error possible between the real value and the polynomial, within the interval from 'a' to 'x'. Calculating this can give you confidence in your approximation, or it might suggest you need to add more terms (a higher order) to get closer to the true value.

Radius of Convergence

The radius of convergence is crucial when working with Taylor series because it tells you where the series works - where it converges to the function you're approximating. Think of it as the 'safe zone' for your series.

In simple terms, it's the distance from the center 'a' beyond which the series does not give an accurate approximation anymore. The series is like a bridge built from 'a' to some point - if built well (the series converges), it'll take you to the correct value. If the series diverges, you'll fall off into the water of mathematical uncertainty.

Finding the Radius:

• Use the ratio test to determine where the series converges.
• Remember, it's all about where the terms of the series stop getting smaller by the ratio you've tested.

The radius of convergence ensures that you're using your Taylor series within a reliable range for the function you’re studying.