Derive the expressions for the mean and variance of a geometric random variable with parameter p. (Formulas for infinite series are required.)
Derive the expressions for the energy and energy-loss curves shown in Figure 3-8 for the damped oscillator. For a lightly damped oscillator, calculate the average rate at which the damped oscillator loses energy (i.e., compute a time average over one cycle).
Derive the expression for the phase paths of the plane pendulum if the total energy is E > 2mgl. Note that this is just the case of a particle moving in a periodic potential U (θ) = mgl (1 – cos θ).
Derive the expression for a de Broglie wavelength λ of a relativistic particle moving with kinetic energy T. At what values of T does the error in determining λ using the non-relativistic formula not exceed 1 % for an electron and a proton?
Derive the disturbance covariance matrix for the model
What parameter is estimated by the regression of the OLS residuals on their lagged values?
Derive a rate expression for each of the following single reactions taking place through a sequence of steps as indicated. Define your rate clearly. S represents an active site.
Derive P3 (x) from the Rodrigues formula, and check that P3 (cos θ) satisfies the angular equation (3.60) for l = 3. Check that P3 and P1 are orthogonal by explicit integration.
Derive (in Cartesian coordinates) the quantum mechanical operators for the three components of angular momentum starting from the classical definition of angular momentum, l = r x p. Show that any two of the components do not mutually commute, and find their commutators.
Derive Guideline 3 in Table 6.10 for a monatomic species of 30 amu:
a) Using collision theory and assuming a 2-dimensional gas.
b) Using absolute rate theory and assuming immobile adsorption.
c) Using absolute rate theory and assuming a 2-dimensional gas.
Derive expressions shear force V, bending moment M and sag w for a â€œweightlessâ€ beam with built-in ends of thickness h, breadth b and span L under a uniform load p. Plot the shear and moment diagrams. Use the sign convention that tension is positive, the x-axis is to the right, the z-axis down, the y-axis is out of the page, a ccw (counterclockwise) moment is positive and a downward shear force is positive as shown infigure.