March 7, 2017

Basic category theory by van Oosten J.

By van Oosten J.

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Working exercise The family of curves called limacons are shown in Fig. 3(d). For C1 , b < a (r ~ 0). For C2 , called a cardioid, b = a (r ~ 0). For C3 , which has an inner and outer loop, b > a (r may take both positive and negative values). Analyse these equations in a similar way to that shown in the previous examples, taking suitable values of a and b. 2 1 The table gives the equations of straight lines and circles in polar and equivalent cartesian forms. Verifythe conversions and sketch the graphs.

By choosing k = 1 we are able to identify a unique value of a, called e, and for this function y = eX and :~ = e", The number e is irrational and its value is 2·71828 . . The function eX is called the exponential function and its graph is shown in Fig. 4(a). Transcendental curves 33 Working exercise On the same axes, sketch the graphs of y = 2\ y = eX and y = 3x • Note that all curves y = a" pass through the point (0, 1). The curve y = 2 X lies below that of y = eX for x > 0 and the curve y = 2X lies above y = eX for x < 0; the curve y = 3X lies above that of y = eX for x > 0 and below for x < O.

4(a» suggests that, for all x E IR, the function is one-one. The graph of the inverse function f- 1 is obtained by reflecting the graph of f in the line y = x , as shown in Fig. 6. Transcendental curves 35 x Fig. 6 The inverse function has domain ~+ and range ~ and is called the logarithm function, written In x or 10& x . The number e is called the base of natural logarithms. y y If y = In x, dd = .! and, for x > 0, dd > 0, confirming, as shown in Fig. 6, that In x is a steadily increasing function.

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