# Calculus: An Intuitive and Physical Approach (2nd Edition) by Morris Kline

By Morris Kline

Application-oriented creation relates the topic as heavily as attainable to technology. In-depth explorations of the by-product, the differentiation and integration of the powers of x, theorems on differentiation and antidifferentiation, the chain rule and examinations of trigonometric features, logarithmic and exponential capabilities, concepts of integration, polar coordinates, even more. Examples. 1967 version. answer consultant to be had upon request.

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Extra resources for Calculus: An Intuitive and Physical Approach (2nd Edition) (Dover Books on Mathematics)

Sample text

2i) i=0 (k−1)/2 i=0 if k is even, t2i+1 (s) f (x), (2i + 1)! 6) if k is odd. P. derivate of f at x of order k + 2 is deﬁned by (s) f (1) (x) = lim sup t→0 (s) f (k+2) (x) = lim sup t→0 1 (f ; x, t). k+2 (f ; x, t), k ≥ 0. P. derivate of f at x of order k + 2, f (s) (x). P. derivative f(k+2) (x), possibly inﬁnite (s) in this case. Note that if f(k+2) (x) is ﬁnite, then this deﬁnition agrees with the one given above; this we now show. 5) f (x + t) + (−1)k f (x − t) 1 2 tk+2 ̟k+2 (f ; x, t) + P (t) (k + 2)!

Tk G(t) = . k! Then since F (0) = F ′ (0) = · · · = F (k−2) (0) = 0 and G(0) = G′ (0) = · · · = G(k−2) (0) = 0 for any t > 0, there is, by the mean value theorem, an ξ such that 0 < ξ < t and F (k−1) (ξ) F (t) = (k−1) . 4) G(t) G (ξ) To see this, for x in some neighbourhood of 0 write k−2 Φ(x) = F (x) + i=1 (t − x)i (i) F (t) G(x) + F (x) − i! G(t) k−2 i=1 (t − x)i (i) G (x) . i! Observe that Φ(0) = Φ(t) = 0 and Φ′ (x) = F (t) (t − x)k−2 (k−1) (t − x)k−2 (k−1) F (x) − G (x); (k − 2)! G(t) (k − 2)!

Let n be a ﬁxed positive even integer, n = 2k say, and consider 0 < h1 < h2 < · · · < hk . 1), then this limit is called the symmetric Riemann∗ derivative of f at x of order n, n = 2k. If n is a positive odd integer, n = 2k − 1 say, consider 0 < h1 < h2 < · · · < hk . 2) hk →0 h1 →0 exists, the last limit being ﬁnite or inﬁnite, while the inner limits are ﬁnite, then this limit is called the symmetric Riemann∗ derivative of f at x of order n, n = 2k − 1. The symmetric Riemann∗ derivative of f at x of order n is ∗(s) written f(n) (x).