Final answer:
The question pertains to representing the function f(x) = πx using a half-range cosine series, which is used for approximating even functions over a specified interval using only cosines. The coefficients of the series are calculated through integrals of the function times cosines over the interval.
Step-by-step explanation:
The question is about using the half-range cosine series to represent the function f(x) = πx over a specified interval. The half-range cosine series is a Fourier series that represents a function using only cosine terms. This is particularly useful for functions that are even symmetrical about the y-axis.
For a function f(x) defined on the interval [0,L], the half-range cosine series is given by:
a0/2 + ∑ (an cos(nπx/L))
where:
- a0 = (1/L) ∫0L f(x) dx
- an = (2/L) ∫0L f(x) cos(nπx/L) dx, for n > 0
The coefficients a0 and an are found through integration. The resulting series will approximate the function f(x) over the interval [0,L]. If the function is defined differently across several intervals, the series must be calculated separately for each interval.