Properties of sin(ax) +c

Problem

Given are the functions \(g_{a,c}: x\mapsto \sin(ax)+c\) defined on \(\mathbb{R}\) with \(a,c\in \mathbb{R}^+_0\).

1. For each of the following two properties find a possible value for \(a\) and a possible value for \(c\) such that the corresponding function \(g_{a,c}\) has this property.

1. The function \(g_{a,c}\) has the codomain \([0;2]\).

2. The function \(g_{a,c}\) contains exactly three zeros in the interval \([0;\pi]\).

2. Determine as a function of \(a\) the possible values of the derivative of \(g_{a,c}\).

Solution of part a

The parameter \(a\) determines the period of the sine but does not influence the function’s codomain. The parameter \(c\), on the other hand, shifts the function’s codomain by a constant value.

1. The codomain of the sine function \(\sin(ax)\) is given by \([-1;1]\). A constant shift by \(c=1\) changes the codomain as specified in the requirement. For the parameter \(a\) an arbitrary nonzero value can be chosen, e.g. \(a=1\). We check our statement with the help of Sage:

2. The number of the function’s zeros in the interval \([0;\pi]\) can be adjusted by means of the parameter \(a\). It is necessary, however, that the codomain of the function includes zero. Choosing \(c=0\), half a period of the sine function with \(a=1\) fits into the given interval, thus leading to only two zeros. In contrast, choosing \(a=2\), a full period of the sine function with exactly three zeros lies within the interval. Again, we check our result with Sage:

Solution of part b

First, we need to determine the derivative of the function \(g_{a, c}(x)\). By means of the chain rule, we obtain

\[\frac{\mathrm{d}g_{a,c}(x)}{\mathrm{d}x} = a\cos(ax).\]

We remark that the derivative is independent of the parameter \(c\). The cosine function possesses the codomain \([-1;1]\) but, in addition, is compressed or stretched because of the amplitude \(a\). The codomain of the derivate thus results in \([-a;a]\).

In order to obtain a general expression for the derivative, in Sage we formally introduce the parameters \(a\) and \(c\) as additional variables of the function. Then, we plot the derivative \(g_{a,c}'(x)\) for a few values of the parameter \(a\). The different amplitudes and periods can easily be read off.