Well this unit is all about Taylor series, and in class he told us to use the Taylor series to get the values of the 40th and 41st derivatives at 0.RUber said:You are evaluating the 40th and 41st derivatives at 0. I am not seeing how this is related to the Taylor series.
Not quite. What's the general formula for the Taylor series?Panphobia said:Homework Statement
If f(x) = x^5*cos(x^6) find f40(0) and f41(0)
The Attempt at a Solution
So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?
Panphobia said:Homework Statement
If f(x) = x^5*cos(x^6) find f40(0) and f41(0)
The Attempt at a Solution
So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?
A Taylor series is a mathematical representation of a function as an infinite sum of terms, with each term corresponding to a derivative of the function evaluated at a single point.
A Taylor series can be used to approximate the value of a function at a certain point, even if the function is not easily calculable at that point. By taking higher order derivatives, the approximation becomes more accurate.
A Taylor series is calculated by taking the derivatives of a function at a specific point, plugging those derivatives into the Taylor series formula, and summing the terms to get the approximate value of the function at that point.
The formula for a Taylor series is: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + f^(n)(a)(x-a)^n/n! where f^(n) represents the nth derivative of the function f at the point a.
The nth derivative plays a crucial role in the accuracy of the Taylor series approximation. The higher the value of n, the more accurate the approximation becomes. However, calculating higher order derivatives can be time-consuming and may not always be necessary for a desired level of accuracy.