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Singularities When a function blows up as the tangent function does at all the odd-integer multiples of p /2, we say that the function is singular for the affected values of the input variable Such a blow-up point is called a singularity If you ve read books or watched movies about space travel and black holes, maybe you ve seen or heard the term space-time singularity That s a place where all the familiar rules of the universe break down In a mathematical singularity, things aren t quite so dramatic, but the output value of a function becomes meaningless In Fig 2-5, the singularities are denoted by vertical dashed lines The dashed lines themselves are known as asymptotes Inflection points Midway between the singularities, the graph of the tangent function crosses the q axis, and the sense of the curvature changes Below the q axis, the curves are always concave to the right and convex to the left Above the q axis, the curves are always concave to the left and convex to the right Whenever we have a point on a curve where the sense of the curvature reverses, we call that point an inflection point or a point of inflection (Some texts spell the word inflexion ) Lots of graphs have inflection points If you re astute, you ll look back in this chapter and notice that the sine and cosine waves also have them From your algebra courses, you might also remember that the graphs of many higher-degree polynomial functions have inflection points

Some students wonder if there s a way to define a function at a singularity If you scrutinize Fig 2-5 closely, you might be tempted to say that tan (p /2) = where the symbol means positive or negative infinity The graph suggests that the output of the tangent function might attain values of infinity at the singular input points, doesn t it It s an interesting notion; the problem is that we don t have a formal definition for infinity as a number Mathematicians have found it difficult, over the generations, to make up a rigorous, workable definition for infinity as a number Some mathematicians have grappled with the notion of infinity and come up with a way of doing arithmetic with it Most notable among these people was Georg Cantor, a German mathematician who lived from 1845 to 1918 He discovered the apparent existence of multiple infinities, which he called transfinite numbers If you re interested in studying transfinite numbers, try searching the Internet using that term as a phrase

Here s a challenge!

Figure out the value of tan (p /4) Don t do any calculations You should be able to infer this on the basis of geometry alone

Draw a diagram of a unit circle, such as the one in Fig 2-1, and place ray OP so that it subtends an angle of p /4 with respect to the x axis (That s exactly northeast if the positive x axis goes east and the positive y axis goes north ) Note that the ray OP also subtends an angle of p /4 with respect to the y axis, because the x and y axes are mutually perpendicular (oriented at an angle of exactly p /2 with respect to each other), and p /4 is half of p /2 Every point on the ray OP is equally distant from the x and y axes, including the point (x0,y0) where the ray intersects the circle It follows that x0 = y0 Neither of them is equal to 0, so you know that y0/x0 = 1 According to the definition of the tangent function, you can conclude that tan (p /4) = y0 /x0 = 1