Obtain the Fourier series of the function defined by f(x) = x², −1

Question

Obtain the Fourier series of the function defined by f(x) = x², −1 <= x <= 1. Obtain the Fourier series of the function defined by f(t) = 1 + t/4, −4 <= t <= 0; 1 − t/4, 0 <= t <= 4 . Sketch f(t) in the interval −8 <= t <= 8. Determine the Fourier series expansion of f(t).

ScanSolve AI Tutor · Accepted Answer

(a) Step 1: The forward difference is defined as Δ f(x) = f(x+1) − f(x). Step 2: For f(x) = x³, first find f(x+1): f(x+1) = (x+1)³. Step 3: Expand (x+1)³ using the binomial theorem: (x+1)³ = x³ + 3x²(1) + 3x(1)² + (1)³ = x³ + 3x² + 3x + 1. Step 4: Substitute into the difference formula: Δ f(x) = (x³ + 3x² + 3x + 1) − x³. Step 5: Simplify: Δ f(x) = 3x² + 3x + 1. Δ f(x) = 3x² + 3x + 1 (b) Step 1: The forward difference is defined as Δ f(x) = f(x+1) − f(x). Step 2: For f(x) = a^(x) where a > 0, a ≠ 1, first find f(x+1): f(x+1) = a^(x+1). Step 3: Rewrite a^(x+1): a^(x+1) = a · a^(x). Step 4: Substitute into the difference formula: Δ f(x) = a · a^(x) − a^(x). Step 5: Factor out the common term a^(x): Δ f(x) = a^(x)(a − 1). Step 6: Note that this can also be written as Δ f(x) = (a−1) f(x), since f(x) = a^(x). Δ f(x) = (a−1)a^(x)

Obtain the Fourier series of the function defined by f(x) = x², -1 <= x <= 1. Obtain the Fourier series of the function defined by f(t) = 1 + t/4, -4 <= t <= 0; 1 - t/4, 0 <= t <= 4 . Sketch f(t) in the interval -8 <= t <= 8. Determine the Fourier series expansion of f(t).

Answers

Step-by-step solution

Part a· 5 steps

The forward difference is defined as

For $f(x) = x^3$ , first find $f(x+1)$

Expand $(x+1)^3$ using the binomial theorem

Substitute into the difference formula

Simplify

Part b· 6 steps
$\Delta f(x) = (a-1)a^{x}$

The forward difference is defined as

For $f(x) = a^{x}$ where $a > 0$ , $a \neq 1$ , first find $f(x+1)$

Rewrite $a^{x+1}$

Substitute into the difference formula

Factor out the common term $a^{x}$

Note that this can also be written as

Try one yourself

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More Mathematics Questions

Obtain the Fourier series of the function defined by f(x) = x², -1 <= x <= 1. Obtain the Fourier series of the function defined by f(t) = 1 + t/4, -4 <= t <= 0; 1 - t/4, 0 <= t <= 4 . Sketch f(t) in the interval -8 <= t <= 8. Determine the Fourier series expansion of f(t).

Answers

Step-by-step solution

Part a· 5 steps

The forward difference is defined as

For f(x)=x3f(x) = x^3f(x)=x3, first find f(x+1)f(x+1)f(x+1)

Expand (x+1)3(x+1)^3(x+1)3 using the binomial theorem

Substitute into the difference formula

Simplify

Part b· 6 stepsΔf(x)=(a−1)ax\Delta f(x) = (a-1)a^{x}Δf(x)=(a−1)ax

The forward difference is defined as

For f(x)=axf(x) = a^{x}f(x)=ax where a>0a > 0a>0, a≠1a \neq 1a=1, first find f(x+1)f(x+1)f(x+1)

Rewrite ax+1a^{x+1}ax+1

Substitute into the difference formula

Factor out the common term axa^{x}ax

Note that this can also be written as

Try one yourself

Still stuck on this one?

More Mathematics Questions

For $f(x) = x^3$ , first find $f(x+1)$

Expand $(x+1)^3$ using the binomial theorem

Part b· 6 steps
$\Delta f(x) = (a-1)a^{x}$

For $f(x) = a^{x}$ where $a > 0$ , $a \neq 1$ , first find $f(x+1)$

Rewrite $a^{x+1}$

Factor out the common term $a^{x}$