March 21, 2007 11:00 am
Here is my attempt to create the graph for function (9) in Spivak, page 42

In[3]:=

Plot[(x + x^2 + x ( Sin[x])^2)/(x Sin[x] + x (Sin[x])^2) , {x, -1, 1}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_2.gif]

The difficult part is determining the proper syntax for the Mathematica expression.
Note that the exponent 2 is now placed outside of a pair of parentheses that clearly specifies what it is that is to be squared. The conventional hand-written expression is potentially ambiguous, but not among mathematicians who are familiar with the conventions of the hand-written expression.
I was surprised at the appearance of the graph. It is much simpler than I would have guessed. I was also expecting some form of wave appearance since the sine function was embedded in the expression.
However the graph does raise the question of whether it is indeed the proper image. How would I know?
I will come back to this question after I obtain graphs for the other equations that Spivak mentions.

Here is the graph for function (10) in Spivak [p. 42]

In[4]:=

Plot[( Sin[x])^2 , {x, -1, 1}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_4.gif]

This looks promising for a wave function. Lets enlarge the domain.

In[5]:=

Plot[( Sin[x])^2 , {x, -5, 5}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_6.gif]

Good.
Now let's try function (11) on page 42

In[6]:=

Plot[Sin[( Sin[x])^2 ], {x, -5, 5}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_8.gif]

Okay. The range is now less than in the previous function.
Now for function (12) on page 43:

In[7]:=

Plot[(( Sin[(Sin[((Sin[x ((Sin[x^2])^2)])^2)])])^2) Sin[(x + Sin[x Sin[x]])/(x + Sin[x])] , {x, -5, 5}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_10.gif]

Wow! Let's try to change the scale of the axes to get a clearer image of what is happening.

In[17]:=

Plot[(( Sin[(Sin[((Sin[x ((Sin[x^2])^2)])^2)])])^2) Sin[(x + Sin[x Sin[x]])/(x + Sin[x])] , {x, -4, 4}, AspectRatio.5, PlotStyleRed] ;

[Graphics:HTMLFiles/index_12.gif]

The graph is not symmetric about the y-axis.
All values of the function are positive.
Does this make sense? This makes sense as the function is basically a sine function squared times a sine function. A sine function squared is always positive.
Is the sine function of the second term always positive? Let's see if we can find out.

In[18]:=

Plot[Sin[((x + Sin[x Sin[x]])/ (x + Sin[x])) ], {x, -5, 5}, PlotStyleRed] ;

[Graphics:HTMLFiles/index_14.gif]

Yes. At least the graph for equation (12) is plausible.
However the fact that the syntax is Mathematica valid, does not necessarily mean that it corresponds to the equation in the book. This can be a critical issue when one is dealing with very complex expressions. One must very carefully take apart each part of the expression and make sure that the syntax is valid for that part. This can be slow, cumbersome, tedious work. But it is essential, particularly when one is relatively new to the notational conventions.

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