s6.pdf - University of Toronto Scarborough Department of Computer Mathematical Sciences MAT A32H Winter 2017 Solutions#6 1 Section 12.1 8 12 16 18 y =

s6.pdf - University of Toronto Scarborough Department of...

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University of Toronto Scarborough Department of Computer & Mathematical Sciences MAT A32H Winter 2017 Solutions #6 1. Section 12.1 8. y = ln( x 2 + 6 x ) , y Chain = Rule 1 x 2 + 6 x ( 2 x + 6) = 2 x + 6 x 2 + 6 x = 2 ( x 3) x ( x 6) . 12. y = x 2 ln x, y product = rule 2 x ln x + x 2 parenleftbigg 1 x parenrightbigg = x (1 + 2 ln x ). 16. f ( w ) = log( w 2 + 2 w + 1) = log 10 ( w 2 + 2 w + 1) , f ( w ) = 1 ln 10 · 1 w 2 + 2 w + 1 · (2 w + 2) = 2 ( w + 1) ln 10 . 18. y = x 2 log 2 x, y product = rule 2 x log 2 x + x 2 x ln 2 = x ln 2 (1 + 2 ln x ). 20. y = x 2 ln x , y quotient = rule 2 x ln x x 2 ( 1 x ) (ln x ) 2 = 2 x ln x x (ln x ) 2 = x (2 ln x 1) ln 2 x . 26. f ( t ) = ln parenleftbigg t 4 1 + 6 t + t 2 parenrightbigg = 4 ln t ln(1 + 6 t + t 2 ) , f ( t ) = 4 t 2 t + 6 1 + 6 t + t 2 . 32. y = ln bracketleftbig (5 x + 2) 4 (8 x 3) 6 bracketrightbig = 4 ln(5 x + 2) + 6 ln(8 x 3) , y = 20 5 x + 2 + 48 8 x 3 . y = 4 (9 + 100) x (5 x + 2) (8 x 3) is also a valid answer. 40. y = ln 2 (2 x +11) = (ln(2 x + 11)) 2 , y = 2 ln(2 x +11) · 2 2 x + 11 = 4 ln(2 x + 11) 2 x + 11 . 46. The slope of the tangent line is y = ln( x ) 1 + x parenleftbigg 1 x parenrightbigg = ln x . At x = 1, y (1) = 1 and the slope is y (1) = 0. Hence the equation of the tangent line to y = x ln x x is y ( 1) = 0 ( x 1) or y = 1, a horizontal tangent. 48. If the demand function is p = 25 ln( q + 2) the revenue function is r = p q = 25 q ln( q + 2) . Hence the marginal revenue is d r d q = 25 ln( q + 2) 25 q q +2 [ln( q + 2)] 2 = 25 ( q + 2) ln( q + 2) q ( q + 2) ln 2 ( q + 2) .
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MATA32H Solutions # 6 page 2 2. Section 12.2 4. y = e 2 x 2 +5 , y = 4 x e 2 x 2 +5 . 6. f ( q ) = e q 3 +6 q 1 , f ( q ) = e q 3 +6 q 1 ( 3 q 2 + 6) = 3 ( q 2 2) e q 3 +6 q 1 . 14. y = e x e x e x + e x , y = ( e x + e x )( e x + e x ) ( e x e x )( e x e x ) ( e x + e x ) 2 = ( e x + e x ) 2 ( e x e x ) 2 ( e x + e x ) 2 = 4 ( e x + e x ) 2 . 16. y = 2 x x 2 , y = 2 x ln(2) x 2 + 2 x (2 x ) = x 2 x ( x ln(2) + 2). 22. f ( z ) = e 1 /z , f ( z ) = e 1 /z parenleftbigg 1 z 2 parenrightbigg = e 1 /z z 2 . 30. f ( x ) = 5 x 2 ln x , f ( x ) Chain = Rule 5 x 2 ln x · ln(5) · parenleftbigg 2 x ln( x ) + x 2 x parenrightbigg = 5 x 2 ln x ln(5) ( x ) (1 + 2 ln x ). Now f (1) = (5 0 )(ln 5)(1)(1 + 0) = ln 5. 36. If the average cost function is c = 850 q + 4000 e (2 q +6) / 800 q , the cost function is c = c q = 850 + 4000 e (2 q +6) / 800 and the marginal cost function is c = 4000 e (2 q +6) / 800 · parenleftbigg 1 400 parenrightbigg = 10 e ( q +3) / 400 .
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