Question

(a) Find the required Fourier series for the given function. (b) Sketch the graph of the function to which the series converges for three periods. (c) Plot one or more partial sums of the series. $$ f(x)=x^{2}-2 x, \quad 0

Short answer

Question: Find the Fourier cosine series for the function f(x) = x^2 - 2x on the range of (0, 4) with a period of 8, and sketch the graph of the function for three periods. Plot partial sums of the series. Answer: The Fourier cosine series for the function f(x) = x^2 - 2x on the given range and period is: $$ F(x) = \frac{2}{3} + \sum_{n=1}^{\infty} \left(\frac{(-1)^n n\pi}{4}+(1-(-1)^n) \frac{4}{n^2 \pi^2}\right)\cos(\frac{n \pi x}{4}) $$ The graph of the function for three periods would show a parabolic shape in the range (0, 4) and the same shape replicated for the adjacent periods (4, 8) and (-4, 0). The partial sum plots depend on the chosen values of n. The more partial sums used, the closer the series converges to the original function in the range (0, 4).

Step-by-step solution

Determine L and the coefficients a_n

To find the Fourier cosine series, we first need to find the value of L, which is half the period. The period is given as 8, so L = 4. Next, we find the coefficients a_n using the integration formulas for a_0 and a_n.

Find a_0

To find a_0, we integrate f(x) over the range of (0, 4) and multiply the result by 1/L: $$ a_0 = \frac{1}{4}\int_{0}^{4} (x^2 - 2x) dx $$ We first solve the integral and then multiply by 1/4: $$ a_0 = \frac{1}{4} \left[\frac{1}{3}x^3 - x^2\right]_0^4 = \frac{1}{4} ( \frac{64}{3} - 16 ) = \frac{4}{3} $$

Find a_n

Now we find a_n using the formula: $$ a_n = \frac{2}{4}\int_{0}^{4} (x^2 - 2x) \cos(\frac{n \pi x}{4}) dx = \frac{1}{2}\int_{0}^{4} x^2 \cos(\frac{n \pi x}{4}) - x \cos(\frac{n \pi x}{4}) dx $$ Using integration by parts twice, we find the final expression for a_n: $$ a_n = \frac{1}{2}\frac{(-1)^n n\pi}{2}+(1-(-1)^n) \frac{4}{n^2 \pi^2} $$

Write the Fourier cosine series

With the expressions for a_0 and a_n, we can write the Fourier cosine series F(x) as: $$ F(x) = \frac{2}{3} + \sum_{n=1}^{\infty} \left(\frac{(-1)^n n\pi}{4}+(1-(-1)^n) \frac{4}{n^2 \pi^2}\right)\cos(\frac{n \pi x}{4}) $$

Sketch the graph of the function for three periods

Now we can sketch the graph of our function f(x) for three periods. Start by plotting the original function for the range (0, 4) then, using the periodicity of 8, replicate the graph for the adjacent periods (4, 8) and (-4, 0).

Plot partial sums of the series

Finally, we can plot one or more partial sums of the Fourier cosine series. A common choice is to plot partial sums for n = 1, n = 2, n = 3, and n = 4. The more partial sums used, the closer the series will converge to the original function in the range (0, 4).

Key concepts

Cosine Series

The cosine series is a special type of Fourier series used to represent even functions in trigonometric terms. When we are dealing with a function like \( f(x) = x^2 - 2x \) over the interval \( 0 < x < 4 \), with period 8, we utilize the cosine series because it focuses entirely on cosine terms. This makes it suitable for functions that are symmetric, as cosine itself is an even function.
To begin, we determine the coefficient \( a_0 \) by integrating the given function over its period and dividing by half the period, also referred to as \( L \). After finding \( a_0 \), we seek the coefficients \( a_n \), using the formula:

• \( a_0 = \frac{1}{L} \int_{0}^{L} f(x) \cos(0) \, dx \)
• \( a_n = \frac{2}{L} \int_{0}^{L} f(x) \cos\left(\frac{n \pi x}{L}\right) \, dx \)

By focusing on cosine terms, this approach simplifies analyzing and graphing functions over symmetric intervals. As a result, the cosine series efficiently captures the essence of the original function across the specified domain and period.

Integration by Parts

Integration by parts is a technique used to solve integrals where the standard methods fail. It is especially useful when dealing with the product of two functions, like polynomial and trigonometric functions. The formula for integration by parts is derived from the product rule of differentiation, and it is given by:

• \( \int u \, dv = uv - \int v \, du \)

Here, you select \( u \) as a function that simplifies or differentiates well, while \( dv \) should be a function whose integral is easily found.
In our original problem, this method helps in dealing with the integral \( \int (x^2 - 2x) \cos\left(\frac{n \pi x}{4}\right) dx \) when computing the coefficients \( a_n \). By performing integration by parts twice, we can tackle complex integrations, transforming them into more manageable forms. This step is essential for simplifying terms and finding the exact coefficients for the function's cosine series.

Graph Sketching

Graph sketching is a crucial part of understanding how a function behaves visually over its specified domain. For the expression \( f(x) = x^2 - 2x \), graphing involves both plotting the original function and understanding its periodic extensions. Start by graphing the basic function on its primary interval from \(0\) to \(4\).
To sketch the graph over three periods, use the periodicity to extend the function graph from \(-4\) to \(8\). This requires you to understand visually how the function repeats itself due to its period of 8. You simply replicate the function from the initial interval \(0\) to \(4\) as there is symmetry every 8 units.
This visualization aids in comprehending how the Fourier series representations approximate different sections of the function across these intervals. Remember, as more terms are used in the Fourier series, the approximation will closely match the original function.

Periodicity in Functions

Periodicity in functions refers to the repeating patterns that occur over regular intervals. For a function with a specific period, such as our function with period 8, this means that the function values repeat every 8 units along the x-axis.
Understanding periodicity is crucial for sketching graphs correctly over multiple intervals and also for organizing the terms in the Fourier series that model these functions.

• A periodic function satisfies \( f(x) = f(x + P) \) for a period \( P \).
• When dealing with periodic functions, the extension of calculations over one period suffices due to their repetitive nature.

By knowing the period, you can simplify many calculations in Fourier analysis, such as finding coefficients and sketching accurate graphs, ensuring they capture the function's entire behavior over its repeating cycle. This repetition allows for efficient use of the Fourier series to map the function onto a broad range of applications.