Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 12 / a / iii

Table of contents
- Solution iii

Solution

The positive-domain graph starts with limiting value

f (0^{+}) = 1

and limiting slope

- 1

, then decreases towards the horizontal asymptote

f = 0

, with

f (q) \sim 1/ (2 q)

for large

q

. In fact direct differentiation gives

f^{'} (q) = - 1 + (1 + \frac{1}{q}) e^{- 1/ q} < 0, f^{''} (q) = \frac{e ^{- 1/ q}}{q ^{3}} > 0.

The first inequality follows from

e^{s} > 1 + s

for

s > 0

. Thus the graph is strictly decreasing and convex upward, with

0 < f (q) < 1

and no stationary points. The integral representation

f (q) = \int_{0}^{1} (1 - e^{- s / q}) d s

also shows its positivity.

Figure 1.
The spherical mass factor $f (q) = 1 - q + q e^{- 1/ q}$ and its two limiting approximations
. Details.

These properties belong to the reusable exponential spherical mass factor.

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