Advanced Math Question:
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Compute the Maclaurin polynomial of degree 4 for the function f(x)=cos(x) In(1-x) for -1<x<1
A Maclaurin series is a Taylor series expansion of a function about 0,
Maclaurin Polynomial And Taylor Polynomial
The Taylor Series, or Taylor Polynomial, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Maclaurin Polynomial, is a special case of the Taylor Polynomial, that uses zero as our single point.
A Maclaurin series is a power series that allows one to calculate an approximation of a function f(x)f(x) for input values close to zero, given that one knows the values of the successive derivatives of the function at zero. In many practical applications, it is equivalent to the function it represents.
An example where the Maclaurin series is useful is the sine function. The definition of the sine function does not allow for an easy method of computing output values for the function at arbitrary input values. On the other hand, it is easy to calculate the values of \sin(x)sin(x) and all of its derivatives when x=0x=0. The Maclaurin series allows one to use these derivative values at zero to calculate precise approximations of \sin(x)sin(x) for inputs close to but not equal to zero. The Maclaurin series is used to create a polynomial that matches the values of \sin(x)sin(x) and a chosen number of its successive derivatives when x=0x=0. The resulting polynomial matches the sine curve closely.
A Maclaurin series can be used to approximate a function, find the antiderivative of a complicated function, or compute an otherwise uncomputable sum. Partial sums of a Maclaurin series provide polynomial approximations for the function.
A Maclaurin series is a special case of a Taylor series, obtained by setting x_0=0x0=0. The Maclaurin series of a function ff is therefore the series