s09_mthsc208_hw19

s09_mthsc208_hw19 - MTHSC 208 (Differential Equations) Dr....

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Unformatted text preview: MTHSC 208 (Differential Equations) Dr. Matthew Macauley HW 19 Due Friday April 3rd, 2009 (1) The function −π ≤ t < π/2, 0 1 −π/2 ≤ t < π/2, f (t) = 0 π/2 ≤ t ≤ π, can be extended to be periodic of period 2π . Sketch the graph of the resulting function, and compute its Fourier series. (2) The function f (t) = |t|, for t ∈ [−π, π ] can be extended to be periodic of period 2π . Sketch the graph of the resulting function, and compute its Fourier series. (3) The function 0 −π ≤ t < 0, t 0 ≤ t ≤ π, can be extended to be periodic of period 2π . Sketch the graph of the resulting function, and compute its Fourier series. f (t) = (4) Consider the 2π -periodic function defined by f (t) = t2 f (t − 2kπ ), −π ≤ t < π, −π + 2kπ ≤ t < π + 2kπ. Sketch this function and compute its Fourier series. (5) Find the Fourier series of the function f (t) = 2 − 3 sin 4t + 5 cos 6t, and sketch the graph of this function (use your calculator). Hint: this problem is simple – don’t do any integrals! (6) Sketch the graph of the function f (t) = sin2 t and find its Fourier series. Hint: Don’t do any integrals! Instead, use a standard trig identity. (7) Which functions from the previous exercises had only cosine terms in their Fourier series expansion? Which functions only had sine terms? Which had both? Do you see a pattern? Hint: compare the symmetries of the graphs of these functions to the symmetries of the graphs of sine waves and cosine waves. ...
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