 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-2/9/b/i/solution

Use the fixed exponents

p, q

in the original PDF. Assume positive

a, b

so arbitrary real powers and a relative change are defined. Set

h = k /100

. With

b

fixed, the exact new-to-old ratio is

(1 + h)^{p}

100 \frac{Δ f}{f} = 100 [p h + \frac{p ( p - 1 )}{2} h^{2} + O (h^{3})] .

Therefore the approximate percentage change through second order is

p k + \frac{p ( p - 1 )}{200} k^{2} .

For negative

k

this same signed expression describes a decrease.

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