morphism said:What you've written down certainly works.
boomer22 said:Homework Statement
All you know about a function f[x] is:
f[0]=2, f '[0]=6, and f ''[0]=-8.
Write down the expansion of f[x] in powers of x through the x^2 term
Homework Equations
none
The Attempt at a Solution
I think the answer is 2 + 6x - 4x^2 , I just don't have any way to check so I want someone to confirm this.
The purpose of finding a series given values of derivatives is to approximate a function using a polynomial series. This allows us to simplify complex functions and make calculations easier. It also helps us understand the behavior of a function at a particular point.
To find a series given values of derivatives, we use the Taylor series expansion. This involves taking the derivatives of a function at a specific point and plugging them into the Taylor series formula. We then use this series to approximate the function at that point.
A Taylor series is a polynomial series that approximates a function at a specific point. A Maclaurin series is a special case of a Taylor series where the point of approximation is 0. This means that all the derivatives of the function are evaluated at 0. In other words, a Maclaurin series is a Taylor series centered at 0.
The accuracy of a series approximation depends on the number of terms used in the series. The more terms we include, the more accurate the approximation will be. However, as we add more terms, the complexity of the calculation also increases. So, there is a trade-off between accuracy and complexity.
No, a series approximation can only be used for functions that are infinitely differentiable at the point of approximation. This means that the function must have derivatives of all orders at that point. Otherwise, the series will not converge and will not accurately approximate the function.