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Math151PopSols

# Math151PopSols - MATH 151 POP QUIZ SOLUTIONS...

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MATH 151 POP QUIZ: SOLUTIONS DERIVATIVES (CHAPTERS 3, 7, and 8) D x x x sin cos ( ) = D x x x cos sin ( ) = - D x x x tan sec ( ) = 2 D x x x cot csc ( ) = - 2 D x x x x sec sec tan ( ) = D x x x x csc csc cot ( ) = - Take a look at each row: we are taking the derivatives of a pair of cofunctions. To get from the first column to the second column, you flip the sign of the result and take the cofunction of each factor. Evaluate the following: D x x x x 4 1 7 3 3 5 - + + Ê Ë Á ˆ ¯ ˜ D x is a linear operator: we can differentiate term-by-term, and constant factors “pop out.” D x x x x x x D x nx x x x x x n n 4 7 4 3 1 3 5 0 12 1 3 5 3 1 3 5 2 2 3 6 1 2 2 3 6 - + + = - - + = = - - - - - - - - ( ) ( ) ( ) ( ) / / / Power Rule: D x x D x x x x x x x 7 3 10 7 3 7 3 10 7 3 14 140 7 3 2 10 2 9 2 2 9 2 9 + = + + = + = + ( ) ( ) ( ) ( ) ( ) "Tail" from the Chain Rule 1 2 44 3 44 D x x x D x x x x x sec sec tan sec tan 4 4 4 4 4 4 4 ( ) [ ] ( ) ( ) ( ) ( ) ( ) = = Tail = 4 12 4 3 4

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D x D x x D x x x D x x x x x x x sin sin sin sin sin cos sin cos 4 4 3 3 3 3 3 3 4 3 3 4 3 3 3 12 3 3 ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) = = = = = Tail 1 2 44 3 44 12 4 3 4 D x x D x x x D x x x x x x x x 3 3 3 2 3 2 3 tan tan tan tan sec ( ) ( ) È Î Í ˘ ˚ ˙ [ ] [ ] ( ) [ ] ( ) = + = + by the Product Rule D x x x D x x D x x x x x x x x x sin sin sin cos sin 3 1 3 1 3 1 3 1 3 1 3 3 1 3 2 2 + = - ¢ = + - + + = + - + Ê Ë Á ˆ ¯ ˜ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = Lo D Hi
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Math151PopSols - MATH 151 POP QUIZ SOLUTIONS...

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