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).

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