# 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**

**Sample text**

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, ...