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Math121SampleQuiz4Sols

Math121SampleQuiz4Sols - QUIZ#4(SAMPLE SOLUTIONS USE A...

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QUIZ #4 (SAMPLE SOLUTIONS) USE A SCIENTIFIC CALCULATOR! 1) Find the derivatives. Simplify where possible. (19 points total) a) D x x ln 3 4 5 + ( ) È Î Í ˘ ˚ ˙ (7 points) Method 1 = + ( ) [ ] = + + ( ) = + = D x x D x x x x 4 5 4 1 5 5 3 3 3 3 2 ln (Using Power Rule for Logs) 6 7 4 8 4 12 5 2 3 x x Method 2 = + ( ) + ( ) [ ] = + ( ) + ( ) ( ) = + ( ) + ( ) = + 1 5 5 1 5 4 5 3 12 5 5 3 4 3 4 3 4 3 3 2 2 3 3 3 4 x D x x x x x x x x 12 5 2 3 x x b) D e x x x 3 4 2 + ( ) (4 points) = + ( ) = + ( ) + ( ) + + + 3 4 4 2 2 e D x x x x x 3 8 1 or 24 3 4 4 2 2 e x e x x x x x
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c) D x e x x 2 6 ( ) (6 points) = ( ) [ ] [ ] + [ ] ( ) [ ] = [ ] [ ] + [ ][ ] = + D x e x D e x e x e x x x x x x 2 6 2 6 6 2 6 2 6 (Product Rule) 2 6 6 2 6 xe x e x x d) D e x 8 ( ) (2 points) = 0 (Hey, we're differentiating a constant….) 2) If f x x x ( ) = ln , find ¢ ( ) f e . Do not approximate. (8 points) ¢ ( ) = ( ) [ ] [ ] + [ ] ( ) [ ] = [ ][ ] + [ ] È Î Í ˘ ˚ ˙ = + ¢ ( ) = + = + = f x D x x x D x x x x x f e e x x ln ln ln ln ln (Product Rule) 1 1 1 1 1 1 2 3) Find the integrals. Simplify wherever possible. (33 points total) a) 4 10 7 3 x x dx - + ( ) Ú (6 points) = Ê Ë Á ˆ ¯ ˜ - Ê Ë Á ˆ ¯ ˜ + + = - + + 4 4 10 2 7 4 2 x x x C x x x C 4 2 5 7 Note: You could have used "Integrating-at-Sight" for the first term.
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