Open the Initial Condition Demo and select any of the curves defined on the interval
[
0
,
𝐿
]
. Now toggle between 'Even' and 'Odd' to extend this curve as an even or odd function, respectively, on the interval
[
−
𝐿
,
𝐿
]
. Which of the following statements is true?

Question

Open the Initial Condition Demo and select any of the curves defined on the interval 
[
0
,
𝐿
]
. Now toggle between 'Even' and 'Odd' to extend this curve as an even or odd function, respectively, on the interval 
[
−
𝐿
,
𝐿
]
. Which of the following statements is true?

BlackTom AI · Accepted Answer

First, let's restate what the problem is asking and what options are provided. The question describes extending a curve defined on [0, L] to [-L, L] by either an even or an odd extension, and asks which statement about the Fourier series representation is true.
Option under consideration: 'Choosing an even function results in cosine terms in the Fourier series representation, but no sine terms.'
Now, evaluate the logic of this statement: When a function is extended evenly, the resulting function is symmetric about x = 0. In Fourier series on the interval [-L, L], even functions have only cosine terms because cosine is an even function and satisfies the symmetry, while sine terms (which are odd) integrate to zero due to the symmetry. Therefore, the Fourier series of an even extension indeed contains only cosine terms and no sine terms.
Next, consider the general complementary case to check for potential confusion: if the original function were extended oddly, the resulting Fourier series would consist exclusively of sine terms (no cosine terms). This serves as a check against the statement and reinforces the idea that the presence of sine terms is tied to odd symmetry, while cosine terms correspond to even symmetry.
What about potential edge cases or common misconceptions? Some might worry about constant (a0) or DC components in the series; those are represented by a0 and a0 multiplies the constant term a0/2 in the cosine-based representation, but this does not introduce sine terms. The key takeaway remains: even extension yields only cosine terms, confirming that sine terms are absent in the Fourier series for an even extension.
In summary, the assertion aligns with Fourier series theory: an even extension on [-L, L] produces a Fourier series consisting solely of cosine terms (and the constant term), with no sine terms included. The statement is consistent with the standard properties of Fourier expansions under even symmetry.
Therefore, from a theoretical standpoint, the claim correctly describes the relationship between even extensions and the absence of sine terms in the Fourier series.

AMME2000 BMET2960 BMET9960 (ND) Week 4 Pre-work

Open the Initial Condition Demo and select any of the curves defined on the interval [ 0 , 𝐿 ] . Now toggle between 'Even' and 'Odd' to extend this curve as an even or odd function, respectively, on the interval [ − 𝐿 , 𝐿 ] . Which of the following statements is true?

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