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 onetoone 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  ex , 2  < x < . (ii) The hyperbolic cosine function is defined as follows: cosh(x) := ex + ex , 2  < x < . 3. (i) Suppose that 1 x 1. We say that sin1 (x) = if /2 /2 and sin() = x. (ii) Suppose that 1 x 1. We say that cos1 (x) = if 0 and cos() = x. (iii) Suppose that  < x < . We say that tan1 (x) = if /2 < < /2 and tan() = x. 4. (Differentiation Theorem for Inverse Functions) Let f be a onetoone 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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 Spring '08
 Skrypka
 Calculus, Inverse function, hyperbolic sine function, interval I. Assume, hyperbolic cosine function, onetoone continuous function

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