Loch ness
Problem
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These following graphs are not monsters. They are humpy because the functions are periodic and involve sines, cosines and absolute values. This problem calls for you to describe and explain the features of the graphs.
- Plot the graph of the function $y=f(x)$ where $f(x) = \sin x +|\sin x|$.
Find the first derivative of this function in the range $0 \le x \le 2 \pi$. Is the first derivative defined everywhere? Can you find an expression for the first derivative for all real $x$?
It might help to start by sketching $y=\sin x$. Can you use this graph to sketch $y=|\sin x|$?
It might help to remember that if $y=x$ then $$\eqalign{ y &= x \ {\rm for } \ x\geq 0 \cr y&= -x \ {\rm for } \ x < 0 .}$$
You may find it useful to split up the domain into regions where $\sin x \ge 0$ and where $\sin x <0$.
See the Getting Started section for some hints on how to show where the derivative is not defined.
- Express the function $f(x) = \sin x + \cos x$ in the form $f(x)=A\sin (x+\alpha)$, find $A$ and $\alpha$ (where $-\pi /2 < \alpha < \pi /2$) and plot the graph of this function.
Similarly express $g(x) = \sin x - \cos x$ in the form $g(x) = B\sin (x +\beta)$, find $B$ and $\beta$ (where $-\pi /2 < \beta < \pi /2$) and plot its graph on the same axes.
We have $A \sin (x + \alpha) = A \sin x \cos \alpha +A \cos x \sin \alpha$, and so equating coefficients gives $$\eqalign{1&=A \cos \alpha \cr 1&=A \sin \alpha}$$
You can then use the identities $\sin^2 \alpha + \cos^2 \alpha = 1$ and $\tan \alpha = \dfrac{\sin \alpha}{\cos \alpha}$ to find $A$ and $\alpha$. You can use a similar idea for the second curve.
- Plot the graph of the function $y=f(x)$ where $f(x)= \sin x + |\cos x|$. Find the first derivative of this function and say where it is defined and where it is not defined.
It might be helpful to consider where $\cos x$ is negative and where it is positive. If we have $\cos x <0$ then $|\cos x| = - \cos x$ (so that we get a positive value).
Getting Started
A smooth graph is one where the first derivative is defined at all points. When you sketch the graph you get one continuous line with no breaks or "kinks" in it.
The modulus function, $f(x)=|x|$, is smooth apart from the point where $x=0$. You can see that there is a "kink" when $x=0$.
As you pass through the point the derivative changes abruptly from $-1$ to $1$. You could write the derivative in this form:
$$\eqalign{ f'(x) &= -1 \ {\rm for } \ x<0 \cr f'(x)&= \phantom{-}1 \ {\rm for } \ x >0 .}$$
Note that $x=0$ is not included in the domain for gradient function above as $f'(x)$ is undefined when $x=0$.
Another way of thinking about where the derivative is defined or not is whether there is a unique tangent to the curve at that point.
The graph below is of the function $y=\text{e}^{-|x|}$. You can see that there is a "kink" when $x=0$.
For most points on this curve there is a unique line that "touches" the curve. At the point $(0,1)$ you can draw many possible lines that touch the curve. There is not a "unique tangent", and the first derivative is not defined at this point.
Student Solutions
Dylan from Brooke Weston in the UK and Norman from Bangkok Patana School in Thailand both completed the problem correctly.
This is Norman's work on the first part of the problem, $f(x)=\sin x + |\sin x|$:
Note that the graph is actually periodic, so there is another 'hump' after $x=2\pi$. This is shown on Dylan's graph:
For the next part of the question, Dylan and Norman both expressed the $f(x)=\sin x + \cos x$ in the form $A\sin{(x+\alpha)}.$ This is Norman's work:
Dylan also expressed $g(x)=\sin x - \cos x$ in this form, and sketched the graphs of $y=f(x)$ and $y=g(x)$ on the same axes:
From here, Dylan and Norman sketched the graph of $y=f(x)=\sin x + |\cos x|$ and calculated the derivative $f'(x).$ This is Norman's work:
Dylan extended $f'(x)$ to include $x\lt-2\pi$ and $x\gt2\pi$:
Teachers' Resources
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 (such as Desmos or Geogebra) 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 $f(x) = A\sin (x + \alpha)$?