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16bf10lec7 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b Di erentiating sin ( t ) and cos ( t ) d dt sin ( t ) = cos ( t ) d dt cos ( t ) =- sin ( t ) Thomas Scanlon Math 16b Why are the stated equations for the derivatives true? We will not give a formal proof, but let us recall what is involved in computing derivatives. d dt ( sin ( t )) = lim → sin ( t + )- sin ( t ) = lim → sin ( t ) cos ( ) + cos ( t ) sin ( )- sin ( t ) = sin ( t ) lim → cos ( )- 1 + cos ( t ) lim → sin ( ) Thomas Scanlon Math 16b Why are the equations true? (continued) For ≈ 0, sin ( ) ≈ 1 and cos ( )- 1 ≈ 0. So, d dt ( sin ( t )) = sin ( t ) lim → cos ( )- 1 + cos ( t ) lim → sin ( ) = cos ( t ) Thomas Scanlon Math 16b Mnemonics for d dt sin ( t ) Thomas Scanlon Math 16b Integrating sin ( t ) and cos ( t ) Reversing the rules of di erentiation we have Z sin ( t ) dt =- cos ( t ) + C Z cos ( t ) dt = sin ( t ) + C Thomas Scanlon Math 16b An example of a volume integral with trigonometric functions...
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16bf10lec7 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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