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

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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 di ff erent 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
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Lecture 7 18.01 Fall 2006 Exam 1 Review General Di ff erentiation Formulas ( u + v ) = u + v ( cu ) = cu ( uv ) = u v + uv (product rule) u = u v - uv (quotient rule) v v 2 d f ( u ( x )) = f ( u ( x )) u ( x ) (chain rule) dx · You can remember the quotient rule by rewriting u = ( uv - 1 ) v and applying the product rule and chain rule.
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