chs5-7review

chs5-7review - Cumulative Review Solutions Chapters 57 1....

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Cumulative Review Solutions 181 Chapters 5–7 1. c. x 2 +16 y 2 =5 x +4 y We differentiate both sides of the equation with respect to x : } d d x } ( x 2 +16 y 2 )= } d d x } (5 x +4 y ) 2 x +32 y } d d y x } =5+4 } d d y x } (32 y –4) } d d y x } =5–2 x } d d y x } = } 3 5 2 y 2 x 4 } . d. 2 x 2 xy +2 y =5 We differentiate both sides of the equation with respect to x : } d d x } (2 x 2 xy +2 y )= } d d x } (5) 4 x 1 y + x } d d y x } 2 +2 } d d y x } =0 (2 – x ) } d d y x } = y –4 x } d d y x } = } y 2 4 x x } . f. ( 2 x +3 y ) 2 =10 We differentiate both sides of the equation with respect to x : } d d x } (2 x +3 y ) 2 = } d d x } (10) 2(2 x +3 y ) 1 2+3 } d d y x } 2 =0 4(2 x +3 y )+ 6(2 x +3 y ) } d d y x } =0 } d d y x } =– } 2 3 } . 2. c. xy 2 + x 2 y =2 at (1, 1) The slope of the tangent line at any point on the curve is given by } d d y x } . We differentiate both sides of the equation with respect to x : (1) y 2 + x 1 2 y } d d y x } 2 +(2 x ) y + x 2 } d d y x } =0 . At (1, 1), 1 + 2 } d d y x } +2+ } d d y x } =0 } d d y x } = –1. An equation of the tangent line at (1, 1) is y –1=–( x – 1) or x + y –2=0 d. y 2 = } 3 7 x x 2 2 + –4 9 } at (1, 2) The slope of the tangent line at any point on the curve is given by } d d y x } . We differentiate both sides of the equation with respect to x : 2 y } d d y x } =. At (1, 2), 4 } d d y x } = } (6)(3) – 3 ( 2 12)(14) } } d d y x } =– } 2 6 5 } . An equation of the tangent line at (1, 2) is y –2=– } 2 6 5 } ( x – 1) or 25 x +6 y –37=0 3. d. f ( x )= x 4 } x 1 4 } f' ( x )=4 x 3 + } x 4 5 } f'' ( x )=12 x 2 } 2 x 0 6 } (6 x )(7 x 2 –4)–(3 x 2 + 9)(14 x ) } (7 x 2 –4)
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4. b. y =( x 2 + 4) (1 – 3 x 3 ) } d d y x } =(2 x )(1–3 x 3 )+( x 2 + 4)(–9 x 2 ) =2 x –6 x 4 –9 x 4 –36 x 2 =2 x –15 x 4 –36 x 2 } d dx 2 y 2 } =2–60 x 3 –72 x 5. s ( t )=3 t 3 – 40.5 t 2 + 162 t for 0 t 8 a. The position of the object at any time t in the interval is s ( t )=3 t 3 – 40.5 t 2 + 162
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This note was uploaded on 02/10/2010 for the course MATHS MCV-01 taught by Professor Mr.m during the Spring '10 term at Seneca.

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chs5-7review - Cumulative Review Solutions Chapters 57 1....

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