practice-quiz5-sol

# practice-quiz5-sol - AMS 151 (Fall, 2009) Joe Mitchell...

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Unformatted text preview: AMS 151 (Fall, 2009) Joe Mitchell Applied Calculus I Practice Problems for Quiz # 5 – Solution Notes 1. Let f ( x ) = 1 − x 3 √ x +2 x . Find f ′ ( x ). We use the quotient rule: f ′ ( x ) = √ x + 2 x · (- 3 x 2 )- (1- x 3 )(1 / 2)( x + 2 x ) − 1 / 2 (1 + (ln 2)2 x ) x + 2 x 2. Let f ( x ) = sin x cos x tan 5 x . Find f ′ ( x ). We use the product rule (twice): write f ( x ) = (sin x ) · (cos x tan 5 x ). f ′ ( x ) = (sin x )(cos x (sec 2 5 x ) · 5 + tan 5 x (- sin x )) + (cos x tan 5 x )(cos x ) 3. Let f ( x ) = e 3 x sin x cos x . Find f ′ ( x ). We use the product rule (twice): write f ( x ) = ( e 3 x ) · (sin x cos x ). f ′ ( x ) = ( e 3 x )(sin x (- sin x ) + cos x (cos x )) + (sin x cos x )( e 3 x · 3) 4. Let f ( x ) = (3 x 4- 2 x 2 + 7) e 3 x . Find f ′ ( x ) and f ′′ ( x ). f ′ ( x ) = (3 x 4- 2 x 2 + 7)( e 3 x · 3) + ( e 3 x )(12 x 3- 4 x ) = (9 x 4 + 12 x 3- 6 x 2- 4 x + 21) e 3 x f ′′ ( x ) = (9 x 4 + 12 x 3- 6 x 2- 4 x + 21)( e 3 x · 3) + ( e 3 x )(36 x 3 + 36 x 2- 12 x- 4) 5. Let f ( x ) = (3 + √ x 9 · 15 x ) 22 . Find f ′ ( x ). f ′ ( x ) = 22(3 + √ x 9 · 15 x ) 21 · (1 / 2)( x 9 · 15 x ) − 1 / 2 · ( x 9 · (ln 15)15 x + 15 x · 9 x 8 ) 6. Let g ( y ) = 7 (13 y +22) . Find g ′ ( y ). g ′ ( y ) = (ln 7)7 (13 y +22) · 13 7. Let f ( x ) = radicalBig x + radicalbig x + √ x . Find f ′ ( x ). We use the chain rule (in succession): f ( x ) = (1 / 2)( x + radicalBig x + √ x ) − 1 / 2 · d dx ( x + radicalBig x + √ x ) = (1 / 2)( x + radicalBig x + √ x ) − 1 / 2 · (1 + (1 / 2)( x + √ x ) − 1 / 2 · (1 + (1 / 2) x − 1 / 2 )) 8. Suppose that h ( b ) = 2 ab + g ( b 2 ) and that g ′ ( w ) = 3 w 2 . Find h ′ ( b ), h ′ ( a ), and h ′ (2). First we compute h ′ ( b ), the derivative of the function h (which is a function of b ; we treat a as constant)....
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## This note was uploaded on 02/07/2010 for the course AMS 151 taught by Professor Zhang during the Fall '08 term at SUNY Stony Brook.

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practice-quiz5-sol - AMS 151 (Fall, 2009) Joe Mitchell...

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