By Colm T. Whelan

The e-book assumes subsequent to no earlier wisdom of the subject. the 1st half introduces the center arithmetic, constantly at the side of the actual context. within the moment a part of the ebook, a sequence of examples showcases a number of the extra conceptually complicated components of physics, the presentation of which attracts at the advancements within the first half. quite a few difficulties is helping scholars to hone their talents in utilizing the offered mathematical tools. suggestions to the issues can be found to teachers on an linked password-protected site for academics.

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**Example text**

34) is the Cauchy form of the remainder term. An alternative form was derived by Lagrange Rn+1 (x) = f (n+1) (????) (x − a)n+1 (n + 1)! 35) with a ≤ ???? ≤ x. 3. If f (x) is a diﬀerentiable function deﬁned on some interval, I, and f ′ (x) = 0 for all x ???? I thenf (x) is constant on I. 34), we have f (x) = f (0) + R1 = f (0) for all x ???? I. 6 Extrema Let us assume that F(x) is a continuous function with a continuous ﬁrst derivate. 37) F(x0 − |h|) − F(x0 ) >0 −|h| We can make |h| arbitrarily small. 38) also holds.

34) is the Cauchy form of the remainder term. An alternative form was derived by Lagrange Rn+1 (x) = f (n+1) (????) (x − a)n+1 (n + 1)! 35) with a ≤ ???? ≤ x. 3. If f (x) is a diﬀerentiable function deﬁned on some interval, I, and f ′ (x) = 0 for all x ???? I thenf (x) is constant on I. 34), we have f (x) = f (0) + R1 = f (0) for all x ???? I. 6 Extrema Let us assume that F(x) is a continuous function with a continuous ﬁrst derivate. 37) F(x0 − |h|) − F(x0 ) >0 −|h| We can make |h| arbitrarily small. 38) also holds.

29) 1! 2! n! where n! = n ⋅ (n − 1) ⋅ (n − 2) · · · 3 ⋅ 2 ⋅ 1 and f (x) = f (a) + x Rn (x) = ∫a f (n+1) (t) (x − t)n ???????? n! 30) Proof: We proceed by induction. 29) is true for n = N, f (x) = f (a) + + f ′ (a) (x − a) 1! x (N+1) f (2) (a) f (N) (a) f (t) (x − a)2 + · · · + (x − a)N + (x − t)N ???????? ∫a 2! N! N! 32) Let u(t) = f N+1 (t) , ???????? = (x − t)N N! then x ∫a f (N+1) (t) (x − t)N N! N! a (N + 1)! Hence, by principle of induction, results true for all n. 33) ◾ Clearly, if Rn goes to zero uniformly as n → ∞, then we can ﬁnd an inﬁnite series.