In mathematics it is often useful to have different ways of viewing
concepts in order to help to build up intuition; however we
need to know to what extent analogies are mathematically
trust-worthy. Imagine the following proposals
concerning calculus -- some might be good analogies, some might
sometimes work and some might simply not work at all.
Consider and test these proposed analogies for understanding
aspects of calculus carefully, applying them to several examples.
Which analogies are largely sound, and which fail to work? Provide
examples of functions to exemplify your points.
Analogy: A curve is a road on a
Imagine an analogy where the curve of a function represents a road
drawn on a map. Imagine driving along this road, starting from the
1. Sign of the derivative of a
function at each point
The derivative of the function is positive when travelling towards
the north, negative when travelling towards the south.
2. Sign of the second derivative
of a function at each point
If your steering wheel is turned clockwise from the neutral
position then the second derivative is negative. If it is turned
anticlockwise from the neutral position then the second derivative
at that point is positive.
3. Sign of the third derivative of
a function at each point
If the steering wheel is in the process of turning in the anti
clockwise direction then the third derivative is positive. If the
steering wheel is in the process of turning in the clockwise
direction then the third derivative is negative.
4. Differentiability condition at
The function is differentiable at points on the road when is it
possible to drive along smoothly without having to suddenly turn
the steering wheel.
5. Points of inflection
Points of inflection occur at the points, and only the points,
where the steering wheel passes through the neutral position.
Note on terminology
'neutral position' is the position of the steering wheel in which
the car travels forwards in a straight line. A clockwise turn
from this position causes the car to turn right and an
anticlockwise turn from this position causes the car to turn