Calculus deconstructed: a second course in first-year - download pdf or read online

Calculus deconstructed: a second course in first-year - download pdf or read online

By Zbigniew H. Nitecki

Calculus Deconstructed is a radical and mathematically rigorous exposition of single-variable calculus for readers with a few past publicity to calculus ideas yet to not equipment of evidence. This e-book is acceptable for a starting Honors Calculus path assuming highschool calculus or a "bridge path" utilizing basica research to encourage and illustrate mathematical rigor. it may function a mix textbook and reference ebook for person self-study. normal issues and strategies in single-variable calculus are offered in context of a coherent logical constitution, development on everyday houses of actual numbers and educating tools of facts via instance alongside the best way. quite a few examples toughen either functional and theoretical knowing, and wide old notes discover the arguments of the originators of the subject.

No earlier event with mathematical facts is believed: rhetorical recommendations and methods of facts (reductio advert absurdum, induction, contrapositives, etc.) are brought by means of instance alongside the best way. among the textual content and routines, proofs can be found for the entire easy result of calculus for features of 1 genuine variable.

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Extra info for Calculus deconstructed: a second course in first-year calculus

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A finite string of digits after the decimal denotes a fraction with denominator a power of 10. d1 d2 d3 . . (where each digit is one of the ten integers {0, 1, . . d1 d2 .. It is clear that this sequence of fractions is non-decreasing (right? 0. The Completeness Axiom now guarantees that this sequence of fractions converges. d1 d2 d3 . . 38 CHAPTER 2. d1 d2 . . dk . We state this formally. 4. Every infinite decimal expression represents a real number, given as the limit of the sequence of finite decimals obtained by truncating the expression.

For example, consider the sequence (−1)kk ∞ k=0 = +1, −1, +1, −1, +1, −1, . . which alternates the two values +1 and −1. This is bounded, since the absolute value of all elements is 1. Nonetheless the sequence diverges. To see this, consider the possibilities. , when k is odd) satisfies (−1)k − 1 = 2 (k odd) so it cannot approximate +1 with accuracy, say, ε = 1. However, if y = 1, we see that every even index term has (−1)k − y = |y − 1| > 0 (k even) and so we can’t approximate y with accuracy, say, half the distance from y to 1: ε = |y − 1| /2.

K = 0, ... 10. αn := sin n, n = 0, ... 11. βn := sin πn , n = 1, ... 15. Cn := k2 k+1 , 1 k sin k, k = 1, ... n i i=0 i+1 , 16. DN := N 2i+1 i=0 3i , 18. Fk := kj=1 k = 1, ... N = 0, ... 19. Gn := nk=0 (−2)−k , n = 0, ... 20. XK := n = 0, ... j j+1 N = 0, ... − K k2 k=0 k+1 , j+1 j , k = 0, ... 21. 05)xm−1 , m = 1, ... 22. x0 = 1, xn := xn−1 xn−1 +1 , n = 1, ... 23. x0 = 2, xn := xn−1 − 1 xn−1 , n = 1, ... 24. x0 = 1, xn := xn−1 + 1 xn−1 , n = 1, ... 25. x0 = x1 = 1, xn = xn−2 + xn−1 for n = 2, ...

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