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Let the student Fig. 7 now show that P = 45° + 0/2 using the fundamental form tan 4, = r/Der. THE DERIVATIVE Sec. 4 51 The Derivative of cos 0. ) The Derivative of sin ke. Consider r = sin ko. Then r + Ar = sin k(6 + A8), and Ar = sin k(8 + A8) - sin k8 = sin k8 [cos I] + cos k8 sin Thus (k A8 (k A8) kB A8 A8 We supply the factor k in the fractions of the right member. AB and 1 Der = 1 0 1 + k cos k8 EXERCISES Using the delta process, find Der and tan ¢. Sketch the ones marked *. *21. r = sin 0 + cos 0 23.

8 41 For example, Y= x2 - 2x x x2 - 4 y 1 Fig. 2 x - 2 Fig. 3 has no point for which has no point for which x=1 x=2 We list here without proof a set of theorems concerning continuity of functions f (x) and g(x) that are necessary for work to follow. If f(x) and g(x) are functions continuous in an interval, then in that interval (1) f(x) + g(x) is continuous (2) f(x) is continuous except where g(x) = 0 g (Z) EXERCISES 1. (a) Given f(x) = x2 + 4x - 2. Find f(-2), f(-1), f(0), f(1/x), fl(x - 1)/xJ.