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 a in the graph of the
function f(x) = Asin(x + a)?