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Questions and Answers
What is the definition of an odd function?
What is the definition of an odd function?
- A function where f(−x) = -f(x) (correct)
- A function where f(−x) = f(x)
- A function that has symmetry around the origin
- A function that has symmetry about the y-axis
Which of the following statements is true about even functions graphically?
Which of the following statements is true about even functions graphically?
- They have no specific symmetry
- They have symmetry about the x-axis
- They have symmetry around the origin
- They have symmetry about the y-axis (correct)
Which of the following functions is considered odd?
Which of the following functions is considered odd?
- $5x^3 - 3x$ (correct)
- $ ext{sin } x$
- $ ext{cos } x$
- $-x^6 + 4x^4 + x^2 - 3$
What can be said about the product of an even function and an odd function?
What can be said about the product of an even function and an odd function?
When integrating an odd function over a symmetric domain, what is the result?
When integrating an odd function over a symmetric domain, what is the result?
What is a characteristic of integrating even functions over symmetric domains?
What is a characteristic of integrating even functions over symmetric domains?
What are the coefficients called in a Fourier series?
What are the coefficients called in a Fourier series?
What is the big advantage of Fourier series compared to Taylor series?
What is the big advantage of Fourier series compared to Taylor series?
What is automatically concluded about the bn coefficients for even functions?
What is automatically concluded about the bn coefficients for even functions?
Which identities are useful for Fourier series when n is an integer?
Which identities are useful for Fourier series when n is an integer?
What should be done to find a Fourier series?
What should be done to find a Fourier series?
What happens to bn if f(x) is an even function?
What happens to bn if f(x) is an even function?
What is the period of a periodic function?
What is the period of a periodic function?
How are sine and cosine functions related to periodicity?
How are sine and cosine functions related to periodicity?
If a function has period 2p, what interval is this function defined on?
If a function has period 2p, what interval is this function defined on?
How can a periodic function be visually explained?
How can a periodic function be visually explained?
What does Fourier series help with in relation to periodic functions?
What does Fourier series help with in relation to periodic functions?
What are the Fourier coefficients a0 and an when the function f(x) is odd?
What are the Fourier coefficients a0 and an when the function f(x) is odd?
If a function is neither even nor odd, which formulas should be used to compute Fourier coefficients?
If a function is neither even nor odd, which formulas should be used to compute Fourier coefficients?
In Example 1, how is the function f(x) defined on the interval [-1,0]?
In Example 1, how is the function f(x) defined on the interval [-1,0]?
What do the Fourier coefficients represent in a Fourier series?
What do the Fourier coefficients represent in a Fourier series?
If a function is periodic and defined on one period, what can be computed using Fourier series?
If a function is periodic and defined on one period, what can be computed using Fourier series?
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Study Notes
Fourier Series
- The Fourier series of a function f(x) is an infinite series involving sines and cosines, represented by the formula: ∞ f (x) = a0 + ∞ [an cos(nπx/p) + bn sin(nπx/p)]
- The Fourier coefficients a0, an, and bn are calculated using the formulas: a0 = (1/p) ∫f(x)dx, an = (1/p) ∫f(x)cos(nπx/p)dx, and bn = (1/p) ∫f(x)sin(nπx/p)dx
Fourier Coefficients
- The Fourier coefficients of an even function simplify to: bn = 0
- The Fourier coefficients of an odd function simplify to: a0 = 0 and an = 0
Even and Odd Functions
- An even function has symmetry about the y-axis, and satisfies the condition: f(-x) = f(x)
- An odd function has symmetry about the origin, and satisfies the condition: f(-x) = -f(x)
- Examples of even functions: sums of even powers of x, cos x
- Examples of odd functions: sums of odd powers of x, sin x
Integrating Even and Odd Functions
- If f(x) is an odd function, then ∫f(x)dx = 0 over a symmetric domain
- If f(x) is an even function, then ∫f(x)dx = 2∫f(x)dx over a symmetric domain
Periodic Functions
- A periodic function has repetitive behavior, and satisfies the condition: f(x + T) = f(x) for every x
- The smallest period T is called the period of the function
- Examples of periodic functions: sin x, cos x, sin(Ï€x), cos(Ï€x)
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