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lec7 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture 7 18.01 Fall 2006 Lecture 7: Continuation and Exam Review Hyperbolic Sine and Cosine Hyperbolic sine (pronounced “sinsh”): sinh( x ) = e x e x 2 Hyperbolic cosine (pronounced “cosh”): e x + e x cosh( x ) = 2 x x d sinh( x ) = d e e x = e ( e x ) = cosh( x ) dx dx 2 2 Likewise, d cosh( x ) = sinh( x ) dx d (Note that this is different from cos( x ).) dx Important identity: cosh 2 ( x ) sinh 2 ( x ) = 1 Proof: 2 x 2 cosh 2 ( x ) sinh 2 ( x ) = e x + 2 e x e 2 e x 1 1 1 cosh 2 ( x ) sinh 2 ( x ) = 4 e 2 x + 2 e x e x + e 2 x 4 e 2 x 2 + e 2 x = 4 (2 + 2) = 1 Why are these functions called “hyperbolic”? Let u = cosh( x ) and v = sinh( x ), then u 2 v 2 = 1 which is the equation of a hyperbola. Regular trig functions are “circular” functions. If u = cos( x ) and v = sin( x ), then u 2 + v 2 = 1 which is the equation
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