note - Part 2: Calculus 1 Chapter 1 Number sets, natural,...

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Part 2: Calculus 1 Chapter 1 Number sets, natural, integer, rational, real, complex. Equation and inequality. Arithmetic and geometric means, triangle inequality. 2 Chapter 2 Sequences, limit of a sequence, ± - N description, examples a n = 2 n - 1 3 n + 2 Infinity, bounded sequence, monotonically bounded sequence Least upper bound and greatest lower bound, suprenum. Cauchy convergence theorem Series, arithmetic series, geometric series, harmonic series, consider se- ries as a sequence, Zeno’s paradox (Achilles and turtle). Fibonacci sequence, golden ratio Theorem of infinite series S n = a 0 + a 1 + ··· + a n lim n →∞ a n +1 a n < 1 the series is convergent. Series S n = X n =1 1 n ( n + 1) Harmonic series diverges. 1
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3 Chapter 3 Functions, bounded and unbounded, monotonic functions. Inverse and implicit functions, multi-valued function. Polynomial, algebraic and transcendental functions. Power function, logarithmic function, trigonometric function. Limit of a function, ± - δ description, left and right limits. Continuity of function, three conditions for continuity, piece-wise con- tinuity. Example: f ( x ) = sin x x Sandwich theorem. The limit of e : e = lim x 0 (1 + x ) 1 x = lim n →∞ ± 1 + 1 n ² n Discuss limits: lim x 0 | x | , lim x 0 | x | x . Examples of limit: lim x 0 x x + 9 - 3 , lim x 0 cos x - 1 x 2 4 Chapter 4 Derivative as rate of change, condition for existence, piecewise differ- entiability. Basic derivatives, rules, example sin x , chain rule, high order deriva- tives. Local extreme values. Example: the longest ladder around corner. 2
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Rolle’s theorem, f ( a ) = f ( b ) = 0 , f 0 ( ξ ) = 0 . mean value theorem, f ( b ) - f ( a ) b - a = f 0 ( ξ ) Inspect F ( x ) = f ( x ) - f ( a ) - ( x - a ) f ( b ) - f ( a ) b - a for Rolle’s theorem. Cauchy’s generalization. f ( b ) - f ( a ) g ( b ) - g ( a ) = f 0 ( ξ ) g 0 ( ξ ) Inspect F ( x ) = f ( x ) - f ( a ) - ( g ( x ) - g ( a )) f ( b ) - f ( a ) g ( b ) - g ( a ) for Rolle’s theorem. L’Hospital’s rules is a result of Cauchy’s generalization. Examples: y = x x , lim x 0 x x . Derivative of inverse function and implicit function. Curvature κ = ds x 2 a 2 + y 2 b 2 = 1 3
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5 Chapter 5 The Newton Leibniz quarrel, who invented integral. Definite integral as a limit of Rieman sum. Integral of basic functions, power, exponential, trignometic, inverse trignometric. Substitution method as an inverse of chain rule, example Z tan x, Z 1 1 - x 2 , Z 1 1 + x 2 Z 1 + cos xdx = Z s 1 - cos 2 x 1 - cos x dx = Z sin x 1 - cos x dx Integration by part as an inverse to the product rule ( uv ) 0 = u 0 v + v 0 u, uv 0 = ( uv ) 0 - u 0 v, Z udv = uv - Z vdu example: Z x cos x, Z xe x , Z e x sin x. Partial fraction, integrable functions
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This note was uploaded on 02/28/2012 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.

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note - Part 2: Calculus 1 Chapter 1 Number sets, natural,...

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