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# defs2 - Sivakumar Some definitions and theorems Exam 2...

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Unformatted text preview: Sivakumar Some definitions and theorems - Exam 2 M151H 1. Suppose that f is a function defined on a domain D. We say that f is one-to-one on D if the following holds: if u, v D and u = v, then f (u) = f (v). 2. (i) The hyperbolic sine function is defined as follows: sinh(x) := ex - e-x , 2 - < x < . (ii) The hyperbolic cosine function is defined as follows: cosh(x) := ex + e-x , 2 - < x < . 3. (i) Suppose that -1 x 1. We say that sin-1 (x) = if -/2 /2 and sin() = x. (ii) Suppose that -1 x 1. We say that cos-1 (x) = if 0 and cos() = x. (iii) Suppose that - < x < . We say that tan-1 (x) = if -/2 < < /2 and tan() = x. 4. (Differentiation Theorem for Inverse Functions) Let f be a one-to-one continuous function on an open interval I. Assume that f is differentiable at the point x0 in I, and that f (x0 ) = 0. Then f -1 is differentiable at y0 = f (x0 ), and (f -1 ) (y0 ) = 1 . f (x0 ) 1 ...
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