Lect17 - 92.131 Lecture 17 1 of 10 Ronald Brent © 2008 All...

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Unformatted text preview: 92.131 Lecture 17 1 of 10 Ronald Brent © 2008 All rights reserved. Derivatives of Trigonometric Functions Let x x f sin ) ( = , then h x h x h x f h x f x f h h sin ) sin( lim ) ( ) ( lim ) ( − + = − + = ′ → → Using x h h x h x cos sin cos sin ) sin( + = + h x x h h x x f h sin cos sin cos sin lim ) ( − + = ′ → ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = ′ → → h x h h x h x x f h h cos sin lim sin cos sin lim ) ( ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ′ → → h h x h h x x f h h sin cos lim ) 1 (cos sin lim ) ( Since x sin , and x cos are constant as far as h is concerned they can be taken out of the limits. 92.131 Lecture 17 2 of 10 Ronald Brent © 2008 All rights reserved. So, ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ′ → → h h x h h x x f h h sin lim cos 1 cos lim sin ) ( Recall we showed: 1 cos lim = − → h h h and 1 sin lim = → h h h in which case, x x x h h x h h x x f h h cos ) 1 ( cos ) ( sin ) sin( lim cos 1 ) cos( lim sin ) ( = ⋅ + ⋅ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⋅ = ′ → → So that x x cos ) (sin = ′ . 92.131 92....
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.

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Lect17 - 92.131 Lecture 17 1 of 10 Ronald Brent © 2008 All...

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