seto6 said:Homework Statement
find the taylor polynomial f4 for f(x)=sin(2x) and a=pi/4
Homework Equations
sin(x)=((-1)^n)(x^(2n+1))/((2n+1)!)
The Attempt at a Solution
so replace x with 2x?
you get ((-1)^n)(2x)^(2n+1)/(2n+1)!)
is this right?
seto6 said:i see i have to center it at pi/4 but this is ((-1)^n)(2x)^(2n+1)/(2n+1)!) centered at 0 so i have to derive it sin(2x) using center pi/4 right
thanks man
The Taylor polynomial is not inherently "bad", but it can sometimes be a poor approximation of a function, especially if the degree of the polynomial is low or the function is highly nonlinear. In these cases, a higher degree polynomial or a different approximation method may be more accurate.
No, the Taylor polynomial can only be used for functions that are infinitely differentiable at a given point. This means that the function must have derivatives of all orders at that point. If the function is not infinitely differentiable, the Taylor polynomial may not be a good approximation.
The accuracy of a Taylor polynomial can be determined by comparing it to the original function. The closer the polynomial is to the function, the more accurate it is. This can be visually assessed by graphing both the polynomial and the function, or by calculating the difference between the values of the polynomial and the function at different points.
Yes, there are alternative methods for approximating functions, such as using other types of polynomials (e.g. Chebyshev polynomials), interpolation methods, or numerical integration techniques. The choice of method depends on the specific function and the desired level of accuracy.
The accuracy of a Taylor polynomial can be improved by increasing the degree of the polynomial, using a smaller interval (i.e. using more data points), or using a different approximation method. It is also important to choose a point of expansion that is close to the point at which the function is being approximated.