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## 'Loch Ness' printed from http://nrich.maths.org/

Why do this problem?

The problem gives a context for investigating the periodic
behaviour of functions involving sines, cosines and the modulus
function and discovering the effects of combining these functions.
Points where the first derivative is not defined occur and are
clearly illustrated by the graph. It is instructive for learners to
realise that although they can find the derivative on both sides of
a point, if it takes different values on each side then the
derivative is undefined and there is no tangent at that
point.

Possible approach

Learners can use graph plotters to plot the graphs and then explain
the form and features of the graph, making the task easier but
perhaps not so rewarding. If they want more of a challenge they can
analyse the equations, sketch the graphs and then use a graph
plotter to check their findings.

Many useful issues for class discussion arise from this problem,
such as how to write down the equation of a function which takes
different values on different intervals, how to interpret the
behaviour of the function where the derivative is undefined, the
amplitude of oscillations etc.

Key questions

When does the sine function take positive values and when is it
negative?

When the derivative of a function at one side of a point has a
different value to the derivative on the other side what happens to
the tangent to a graph at that point?

What is the significance of $A$ and $\alpha$ in the graph of the
function $f(x) = A\sin (x + \alpha)$?