Fourier Series · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Repeating waves built from pure tones

Mathematics · Differential Equations · ages 20-21
Name ______________________   Date ____________

  1. Because sin(mx) and cos(nx) always integrate to zero against each other over one full period, multiplying a Fourier series by cos(nx) and integrating isolates just the an term.

    Circle one:   True   False

  2. The square wave above is odd. What does that say about its cosine coefficients?

    • Every an is 0
    • Every bn is 0
    • a0 is infinite

  3. The sawtooth f of x equals x on minus pi to pi has b2 equal to minus 1. Type b2.

    Answer: ______________

  4. For that square wave, what is b1, the coefficient of sine x?

    • 2 over pi
    • 0
    • 4 over pi

  5. A function is odd (rotationally symmetric through the origin). Which coefficients must be zero in its Fourier series?

    • a0 and all an
    • all bn
    • only a0
    • none of them

  6. For f(x) = x on (0, π), expanded as a half-range cosine series, what is a0?

    • π/2
    • π
    • 2π
    • π²/2

  7. If you multiply a periodic function's Fourier series by sin(3x) and integrate over one full period, which coefficient does that isolate?

    • a0
    • a3
    • b3
    • b1

  8. A sawtooth wave is f(x) = x on (-π, π), repeated with period 2π. What is b2, the coefficient of sin(2x)?

    Answer: ______________

  9. Near the square wave jump, the partial sums overshoot no matter how many terms you keep. What should you conclude?

    • The coefficients were computed wrongly
    • Overshoot near jumps is built into the method
    • Only even functions overshoot

  10. A circuit is driven by a square wave. How does the series help solve it?

    • It replaces the circuit with a resistor
    • It splits the messy drive into pure tones solved one by one
    • It removes the need for any differential equation
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Repeating waves built from pure tones W1-mt_DJ3iwfj7NK-s1

  1. True · This is exactly why the coefficient formulas work: orthogonality wipes out every term except the one you multiplied by.
  2. Every an is 0 · Odd functions are built from sines alone, so no cosine survives.
  3. -1 · The odd sawtooth keeps sines only, and the sine 2 x coefficient is minus 1.
  4. 4 over pi · Splitting the integral where the wave switches sign gives 4 over pi.
  5. a0 and all an · Cosine is even and an odd function times an even function integrates to zero over a symmetric interval, so a0 and every an vanish.
  6. π · a0 is twice the average value of f over (0, π), which comes out to π.
  7. b3 · Every cosine term and every other sine term vanishes against sin(3x), leaving only b3.
  8. -1 · f(x) = x is odd, so only sine terms show up, and the formula bn = 2(-1)^(n+1)/n gives b2 = -1.
  9. Overshoot near jumps is built into the method · The series struggles at discontinuities, and extra terms never fully fix the jump.
  10. It splits the messy drive into pure tones solved one by one · Each tone drives its own easy single frequency problem, and the answers add up.
Worksheet · LightMySky