You are supposed to write f(x) = x2 + 3x - 5 as a power series in powers of (x - 4) instead of in powers of x. Since the function is of degree two, your Taylor's series will also be of degree two. IOW, it will be f(x) = a0 + a1(x - 4) + a2(x - 4)2.lmannoia said:Homework Statement
Let f(x)=x2 +3x -5, and let the summation (from n=0 to infinity) an (x-4)n be the Taylor series of f about 4. Find the values of a0, a1, a2, a3, and a4.
Homework Equations
The Attempt at a Solution
What am I supposed to do with the summation? And what does it mean to find a Taylor polynomial 'about 4'? I'm confused on how to relate both of those things to the f(x) that they gave me.
A Taylor polynomial is a mathematical function used to approximate a more complex function. It is composed of a finite number of terms, each of which is a polynomial of increasing degree.
The coefficients a0, a1, a2, a3, and a4 can be found by using the Taylor series formula, which involves taking derivatives of the original function at a specific point, and plugging in those values into the formula. The coefficients are the values of the derivatives at that specific point.
The coefficient a0 represents the value of the original function at the point of approximation. The coefficient a1 represents the slope of the original function at that point. The coefficient a2 represents the concavity of the original function at that point, and so on for higher degree coefficients. These coefficients help to accurately approximate the original function within a certain range.
The higher the degree of the Taylor polynomial, the more accurate the approximation will be. This is because a higher degree polynomial will have more terms, allowing it to capture more details and nuances of the original function.
Yes, Taylor polynomials can be used for any type of function, as long as the function is smooth and has well-defined derivatives at the point of approximation. However, the accuracy of the approximation may vary depending on the complexity of the function and the degree of the Taylor polynomial used.