Open Educational Resources

Numerical Differentiation: Problems

  1. Compare the basic formulas and the high-accuracy formulas to calculate the first and second derivatives of e^{x} at x=2 with h=0.1. Clearly indicate the order of the error term in the approximation used. Calculate the relative error E_r in each case.
  2. A plane is being tracked by radar, and data are taken every two seconds in polar coordinates \theta and r.
    radar

    Time t (sec.) 200 202 204 206 208 210
    \theta (rad) 0.75 0.72 0.70 0.68 0.67 0.66
    r (m.) 5120 5370 5560 5800 6030 6240

    The vector expression in polar coordinates for the velocity v and the acceleration a are given by:

        \[\begin{split} v&=\frac{\mathrm{d}r}{\mathrm {d}t}e_r+r\frac{\mathrm{d}\theta}{\mathrm {d}t}e_{\theta} \\ a&=\left(\frac{\mathrm{d}^2r}{\mathrm {d}t^2}-r\left(\frac{\mathrm{d}\theta}{\mathrm {d}t}\right)^2\right)e_r+\left(r\frac{\mathrm{d}^2\theta}{\mathrm {d}t^2}+2\frac{\mathrm{d}r}{\mathrm {d}t}\frac{\mathrm{d}\theta}{\mathrm {d}t}\right)e_{\theta} \end{split} \]

    Use the centred finite difference basic formulas to find the velocity and acceleration vectors at t = 206 sec. as a function of the unit vectors e_r and e_{\theta}. Then, using the norm of the velocity and acceleration vectors describe the plane speed and the magnitude of its acceleration.

  3. Fick’s first diffusion law states that:

        \[ \mbox{Mass Flux}=-D\frac{\mathrm{d}C}{\mathrm{d}x} \]

    where \mbox{Mass Flux} is the mass transported per unit area and per unit time with units g/m^2/s, D is a constant called the diffusion coefficient given in m^2/sec., C is the concentration given in g/m^3 and x is the distance in m. The law states that the mass transported per unit area and per unit time is directly proportional to the gradient of the concentration (with a negative constant of proportionality). Stated differently, it means that the molecules tend to move towards the area of less concentration. If the concentration of a pollutant in the pore waters of sediments underlying the lake is constant throughout the year and is measured to be:

    x (m) 0 1 3
    C, 10^{-6} (g/m^3) 0.06 0.32 0.6

    where x=0 is the interface between the lake and the sediment. By fitting a parabola to the data, estimate \frac{\mathrm{d}C}{\mathrm{d}x} at x=0. If the area of interface between the lake and the sediment is 3.6\times 10^6 m^2, and if D=1.52\times 10^{-6}m^2/sec, how much pollutant in g would be transported into the lake over a period of one year?

  4. The positions in m. of the left and right feet of two squash players (Ramy Ashour and Cameron Pilley) during an 8 second segment of their match in the North American Open in February 2013 are given in the excel sheet provided here. Using a centred finite difference scheme, compare the average and top speed and acceleration between the two players during that segment of the match. Similarly, using a parabolic spline interpolation function, compare the average and top speed and acceleration between the two players. Finally, compare with the average speed of walking for humans.

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