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Unformatted text preview: 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 lecturerincharge, 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 MATH1151 Student Support Calculus Notes • Notation N = { , 1 , 2 , ... } is the set of natural numbers. Z = { ..., 2 , 1 , , 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 nonempty 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 xvalue), you get back a real number (the yvalue). onetoone 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 yvalue from the codomain ( R ), say y = 2 for example, and there is no xvalue 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 yvalue from the codomain ( R + ), we can find a corresponding xvalue from the domain ( R ) that satifies e x = y . If a function is onetoone 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....
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