By Vladimir Ivanovic Smirnov

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**Example text**

We can get another antiderivative by adding any arbitrary constant. It is, however, customary to leave out this arbitrary constant, as long as we are only trying to ﬁnd one antiderivative. We shall therefore in this chapter always understand that a constant should be added in the end, and never write it. When it comes to the applications like in diﬀerential equations in Chapter 5 we must never forget this extra arbitrary constant. 2 Figure 24: The graph of f (x) = √ 1 − x2 , x ∈ [−1, 1]. Warning.

All the strange looking manoeuvres above are necessary, because of what we mentioned vaguely in the introduction, namely that diﬀerentiation introduced by the deﬁnition (51) has some very bad properties. The pedagogical problem at this stage of the development is that since one in the elementary Calculus usually starts with working in spaces of the type C ∞ (I0 ) (in fact in the smaller class of analytical functions which behave very nicely) where the diﬀerentiation given by (51) is straightforward, the student does not immediately see why the tedious accuracy above really is necessary.

They are analogous to the trigonometric functions and they share many of their properties, but here we can in their deﬁnitions refer directly to the real exponential function. com 31 The Elementary Functions Calculus 1a (33) cos x = � 1 � ix e + e −i x , 2 sin x = It follows from (32) and (33) that � 1 � ix e + e−i x = cos x, cosh(i x) = 2 � 1 � ix e − e−i x . 2i and sinh(i z) = � 1 � ix e − e−i x = i sin x, 2 which show the close relationship between the trigonometric functions and the hyperbolic functions.