By van Oosten J.
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Make math a snap with ALGEBRA for students. utilizing daily language and many examples, Kaufman and Schwitters help you observe algebra thoughts and ace the attempt. This quantity additionally comes with Interactive Skillbuilder CD-ROM. This application is jam-packed with over eight hours of video guideline to aid all of it make experience.
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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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