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MATH1151 Calculus Notes - DISCLAIMER Although we attempt to...

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DISCLAIMER Although we attempt to be as accurate as possible with our support materials, mistakes will appear from time to time, since these documents are being published by students. ASOC accepts no responsibility for any damage (academic or otherwise) that occurs as a result of any error within these support materials. For official advice regarding your university courses, seek guidance from the relevant lecturer-in-charge, or your tutor. If you are unsure about anything, it is best to seek counsel from the appropriate academics within the university. We would appreciate it if you point out any mistakes by contacting us - [email protected] 1
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MATH1151 Student Support Calculus Notes Notation - N = { 0 , 1 , 2 , . . . } is the set of natural numbers. - Z = { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } is the set of integers. - Q = n p q : p, q Z , q 6 = 0 o is the set of rational numbers. - p Z means p is an element of Z . i.e. p is an integer. - R is the set of real numbers. - means ‘for all’ - means ‘there exists’ - x [0 , ) means 0 x < , whereas x (5 , 26] means 5 < x 26. - a b means a implies b . - a b means a iff (if and only if) b . i.e. a b AND b a The Exponential Function The Least Upper Bound Axiom - Every non-empty set of real numbers that has an upper bound has a LEAST upper bound. The Exponential Function - lim n →∞ 1 + x n n = e x for any x R . - f : N |{z} domain R |{z} codomain If you input a value from the domain, you get back a value from the codomain. In this example, if you input a natural number (the x -value), you get back a real number (the y -value). - one-to-one means f ( x ) = f ( y ) if and only if x = y . - onto means that if y codomain then x domain such that f ( x ) = y . e.g. Consider g : R R , defined by g ( x ) = e x . This function is not onto, because we can take a y -value from the codomain ( R ), say y = - 2 for example, and there is no x -value from the domain ( R ) that satisfies e x = - 2. In contrast, if we have h : R R + , defined by h ( x ) = e x , this function is onto, because for any y -value from the codomain ( R + ), we can find a corresponding x -value from the domain ( R ) that satifies e x = y . - If a function is one-to-one and onto, it is invertible. Hyperbolic and Inverse Hyperbolic Functions - cosh x = e x + e - x 2 and sinh x = e x - e - x 2 - cosh is an even function, while sinh and tanh are odd functions. - tanh x = sinh x cosh x , coth x = cosh x sinh x , sech x = 1 cosh x , cosech x = 1 sinh x - Osborne’s Rule Replace trig functions with hyperbolic functions, and change the sign of a product of sines or implied product of sines (e.g. tan 2 x ). e.g. cos 2 x + sin 2 x = 1 becomes cosh 2 x - sinh 2 x = 1. - y = sinh - 1 x means x = sinh y for any x R - y = cosh - 1 x means x = cosh y for x 1 and y 0. - y = sinh - 1 x = ln( x + x 2 + 1) x R - y = cosh - 1 x = ln( x + x 2 - 1) x 1 - y = tanh - 1 x = 1 2 ln 1 + x 1 - x x ( - 1 , 1) Version 3, 5/6/08 2
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MATH1151 Student Support Calculus Notes Limits Formal Definition - ‘ lim x a f ( x ) = l ’ means: ‘For all real numbers > 0, there exists a real number δ > 0, such that: | f ( x ) - l | < whenever 0 < | x - a | < δ ’.
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