Math16A - Fall 05 Final Solutionss

Math16A Fall 05 - SOLUTIONS TO MATH 16A AUTUMN 2005 FINAL 1 Use logarithmic differentiation to compute f x/f x = d dx(ln f x = d dx(ln x 1 1 x = d

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Unformatted text preview: SOLUTIONS TO MATH 16A, AUTUMN 2005 FINAL 1. Use logarithmic differentiation to compute f ( x ) /f ( x ) = d dx (ln( f ( x ))) = d dx (ln(( x + 1) 1- x )) = d dx ((1- x )ln( x + 1)) =- ln( x + 1) + 1- x x + 1 Thus, f (2) = f (2)(- ln(2 + 1) + 1- 2 2+1 ) = 3- 1 (- ln(3) + (- 1 / 3)) =- ln(3) 3- 1 9 . 2. a. A, b. G, c. E, d. B, e. H 3. Write f ( t ) = # of bacteria in tens of thousands t hours after noon . We know that f is an exponential function. So, we may write f ( t ) = Ae rt for some constants A and r . We know 3 = f (0) = A and 4 = f (1) = 3 e r so that e r = 4 / 3. Hence, g (3) = 3 e r 3 = 3(4 / 3) 3 = 3(64 / 27) = (64 / 9). That is, there are (64 / 9) × 10 4 = 71 , 1111 / 9 individuals at 3pm. bf 4. Z 2 1 (2 x 3- 5 e- 7 x + 4 √ x- 8 x ) dx = [ 1 2 x 4 + 5 7 e- 7 x + 4 5 x 5 4- 8ln( x )] | 2 1 = (8 + 5 7 e- 14 + 4 5 2 5 4- 8ln(2))- ( 1 2 + 5 7 e- 7 + 4 5- 8ln(1)) = 8 + 5 7 e- 14 + 8 5 4 √ 2- 8ln(2)- 1 2- 5 7 e- 7- 4 5 = 6 . 7 + 5 7 e- 14 + 8 5 4 √ 2- 8ln(2)- 5 7 e- 7 5. a. J, b. B, c. D, d. J, e. Fa....
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This note was uploaded on 05/12/2010 for the course CHEM 3A taught by Professor Fretchet during the Fall '08 term at University of California, Berkeley.

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Math16A Fall 05 - SOLUTIONS TO MATH 16A AUTUMN 2005 FINAL 1 Use logarithmic differentiation to compute f x/f x = d dx(ln f x = d dx(ln x 1 1 x = d

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