Solutions Homework 6

Solutions Homework 6 - MATH 106 CALCULUS I FOR BIO. &...

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Unformatted text preview: MATH 106 CALCULUS I FOR BIO. & SOC. SCI. FALL 2011 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Solutions Homework 6. Please show all your work. (1) (5 Points) Suppose that f and g are differentiable functions such that f (1) = 2 , f (2) = 1 , f ′ (1) = − 2 , f ′ (2) = − 1 g (1) = − 2 , g (2) = 2 , g ′ (1) = 3 , g ′ (2) = 1 . (a) If a ( x ) = f ( √ x ) compute a ′ (1). Solution: a ′ ( x ) = f ′ ( √ x ) parenleftbigg 1 2 √ x parenrightbigg = f ′ ( √ x ) 2 √ x When x = 1 we get a ′ (1) = f ′ (1) 2 √ 1 = − 2 2 = − 1 . (b) If b ( x ) = radicalbig f ( x ) compute b ′ (1). Solution: Note that b ( x ) = ( f ( x )) 1 / 2 . b ′ ( x ) = 1 2 ( f ( x )) − 1 / 2 f ′ ( x ) = f ′ ( x ) 2 radicalbig f ( x ) . When x = 1 we get b ′ (1) = f ′ (1) 2 radicalbig f (1) = − 2 2 √ 2 = − 1 √ 2 . (c) If c ( x ) = f ◦ f ( x ) compute c ′ (1). Solution: c ′ ( x ) = f ′ ( f ( x )) f ′ ( x ) Therefore c ′ (1) = f ′ ( f (1)) f ′ (1) = f ′ (2) f ′ (1) = ( − 1)( − 2) = 2 . (d) If d ( x ) = f ◦ g ( x ) compute d ′ (2). Solution: d ′ ( x ) = f ′ ( g ( x )) g ′ ( x ) . Thus d ′ (2) = f ′ ( g (2)) g ′ (2) = f ′ (2) g ′ (2) = ( − 1)(1) = − 1 . 1 2 INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ (e) If e ( x ) = f ( x ) g (2 x ) compute e ′ (1). Solution: e ′ ( x ) = g (2 x ) f ′ ( x ) − f ( x )(2 g ′ (2 x )) ( g (2 x )) 2 ....
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This note was uploaded on 01/20/2012 for the course CALC AS.110.106 taught by Professor Josegomez during the Fall '11 term at Johns Hopkins.

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Solutions Homework 6 - MATH 106 CALCULUS I FOR BIO. &...

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