Extra Problems

  1. The table below contains data from a freshman lab experiment measuring g, the acceleration due to gravity. The experiment entails dropping an object from a height of 1 meter and checking its position every 50 milliseconds using a photogate. Use a least squares fit to estimate g from this data.

    Time and Distance
    t (ms) 0  50  100  150  200  250
    y (cm) 100  97   96  90  78  71

  2. Use the Gram-Schmidt process to turn {1,x,x2} into an orthogonal basis for R2[x] in the inner product
    < f | g > = &int01 f(x)g(x)dx.

    Answer: {1, x − 1/2, x2 − x + 1/6}

  3. Use the orthogonal basis above to obtain the quadratic polynomial that gives the best continuous least-squares fit for the function f(x) = sin(πx) on the interval [0,1]. Use the same inner product as in the previous problem.

  4. Derive the orthogonality relations < e ijπt/L | e ikπt/L > = 2Lδj,k.

  5. Find the complex Fourier series for f(x) = x on [0, 2L].

    Answer:

  6. Using the inner product < f | g > = &int02L f(x) g(x)dx, find the least squares fit to f(x) = x on [0, 2L] from span{1,eiπx/L, e−iπx/L}.

    Answer:

    L + (iL/π)eiπx/L - (iL/π)e−iπx/L