Examples on Partial Differentiation

Comp.145  February 1997  Dr. A.J. Mestel

Assessed work; to be handed in to the general office by Wednesday 12th February. Put my name and NOT that of your PMT on the folder.

At the top of your script write your name and date of birth. In what follows, k is the number of letters in your last name, while a and b are the day and month of your birthday, so that if your name is Mestel and you were born on 13th March, k = 6, a = 13 and b = 3.

1. The price of hiring a personalised computer, P, is a function only of the CPU required, c, and the amount of storage used, s, according to the formula

P (c, s) = kc^{a}s^{b}

when c, s and P are measured in appropriate units. If c and s depend on a parameter N, derive a relation between

\frac{1}{P} \frac{dP}{dN} ,  \frac{1}{c} \frac{dc}{dN}  and \frac{1}{s} \frac{ds}{dN} .

Would it be cheaper to use 1% less CPU time if it required a 4% increase in storage?

Now suppose that I want to run a program involving large N × N matrices, performing operations such as matrix multiplications, Gaussian elimination and so on. If c = αN^{p} and s = βN^{q} for constants α and β, what values would you expect for p and q?

Using these values, calculate \frac{dP}{dN} when P = 10^{3} and N = 10^{4}.

2. Find the stationary point(s) of the function

f(x, y) = \frac{1}{2}(a − b)(x^{2} + y^{2}) + (a + b)xy .

(Recall that a and b are determined by your birthday.) Find the eigenvalues of the Hessian matrix at any stationary point, and hence determine its nature.