lec7 - Lecture 7 18.01 Fall 2006 Lecture 7: Continuation...

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Unformatted text preview: 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 of a circle. 1 Lecture 7 18.01 Fall 2006 Exam 1 Review General Differentiation Formulas ( u + v ) = u + v ( cu ) = cu ( uv )...
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

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lec7 - Lecture 7 18.01 Fall 2006 Lecture 7: Continuation...

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