Question

Find the coefficients of the Fourier cosine series f(x) = 40 + Σ Ak cos(mkπ) for the function defined by:

f(x) =
- 1 if 0 < x < 3
- e^x if x is in (0, 3)
- 5 if x is in (4, [infinity])

Use pi for π. Give the general form of the coefficient Bk.

Answer 1

To find the coefficients of the Fourier cosine series for the given function, we need to determine the values of Ak for each mkπ term in the series. We find the coefficients using integral calculations for the different intervals of the function. The general form of the coefficient Bk for the Fourier cosine series is Bk = 0 for all k, since the function is even.

Step-by-step explanation:

To find the coefficients of the Fourier cosine series for the given function, we need to determine the values of Ak for each mkπ term in the series.

1. For the interval (0, 3), the function is -1 - e^x. To find A0, we need to find the average value of the function over the interval (0, 3) using the formula A0 = (1/3) * ∫((-1 - e^x), dx, 0, 3).

2. For the interval (4, ∞), the function is -5. To find the other coefficients, we need to use the formula Ak = (2/L) * ∫(f(x) * cos(mkπx/L), dx, 4, ∞), where L is the length of the interval (4, ∞) and m is the order of the term.

3. The general form of the coefficient Bk for the Fourier cosine series is Bk = 0 for all k, since the function is even.