Sounds like you should start by using Taylor's formula on the two expressions [itex]f(x_0+h)[/itex] and [itex]f(x_0+2h)[/itex].Ryuuken said:I'm not sure where to start.
A Taylor polynomial is a mathematical approximation of a function using a finite number of terms from its power series expansion. It is a useful tool for approximating complex functions, especially when the function is difficult to evaluate directly.
Expanding a Taylor polynomial allows us to approximate a function with a polynomial of a finite degree, which is often easier to work with than the original function. This can be useful in various areas of mathematics, physics, and engineering.
A Taylor series is an infinite sum of terms from the power series expansion of a function, while a Taylor polynomial is a finite sum of these terms. The more terms we include in a Taylor polynomial, the closer it approximates the original function, but it will never be an exact representation like a Taylor series.
The coefficients of a Taylor polynomial can be found using the formula for the nth derivative of a function evaluated at a specific point. These coefficients can also be calculated using the Remainder or Lagrange form of the Taylor polynomial.
The remainder term in a Taylor polynomial represents the difference between the original function and its approximation using the polynomial. It is important because it allows us to estimate the error in our approximation and improve the accuracy of the polynomial by including more terms.