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Additional resources for Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus
C 1 functions are also called continuously diﬀerentiable. Example 107 All polynomial functions as well as exp, sin, and cos are C ∞ functions. The function ⎧ ⎨ x2 f (x) = ⎩−x 2 if x ≥ 0, if x < 0 is continuous on R. Its derivative exists, is continuous, and is deﬁned by ⎧ ⎨ 2x if x ≥ 0, f (x) = ⎩−2x if x < 0. The second derivative f , however, does not exist at x = 0, where there is a jump from −2 to 2. Therefore f is in C 1 , but not in C 2 . Let us close with the very important mean value theorem.
Example 109 We look at the ﬁrst Taylor polynomials in 0 of the function f (x) = cos(x) + sin(2x). Derivatives of f must be calculated ﬁrst: D 0 f (x) = f (x), D 1 f (x) = − sin(x) + 2 cos(2x), D 2 f (x) = − cos(x) − 4 sin(2x), D 3 f (x) = sin(x) − 8 cos(2x), D 4 f (x) = cos(x) + 16 sin(2x). For the Taylor expansion of f in 0, these derivatives must be evaluated at 0: D 0 f (0) = 1, D 1 f (0) = 2, D 2 f (0) = −1, D 3 f (0) = −8, D 4 f (0) = 1. 3 Taylor’s Formula Taylor 00 f (x) = t0 = Taylor 10 f (x) = t1 = Taylor 20 f (x) = t2 = Taylor 30 f (x) = t3 = Taylor 40 f (x) = t4 = 57 1 , 0!
Proof The absolute convergence follows immediately from the ratio zk+1 (k+1)! zk (k)! = z , k+1 which tends to 0 for k → ∞, and the proposition 252 applies. 1 Fundamental Properties of the Exponential Function In this subsection, we want to deal with some technical aspects which are of general interest, but which are also crucial for the establishment of fundamental properties of the exponential function. In particular, we want to calculate the value exp(w +z), and since this involves the powers (w + z)k as functions of w and z, we need to calculate polynomials (X + Y )k ∈ Z[X, Y ] ﬁrst.