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Math1011_Week09_Prelim2B

# Math1011_Week09_Prelim2B - Math1011 Learning Strategies...

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Math1011 - Learning Strategies Center Various Prelims II-B1 2 3 The equation x 11 defines the following curve: xy y + + = Find the equation of the tangent line at (1, 2). dy dx Find if sin 1 x x y ye + =

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Math1011 Various Prelims B1 2 3 The equation x 11 defines the following curve: xy y + + = Find the equation of the tangent line at (1, 2). 2 2 2 3 d d d d dx dx dx dx 2 d dx 2 2 2 3 2(1) (2) 4 1 (1) 3(2) Implicit Differentiation: (x ) ( ) ( ) (11) 2 ' 3 ' 0 Note: Use product rule for ( ) xy'+3y ' 2 '( 3 ) 2 ' at point (1, 2): ' x y x y xy y x xy y y y xy y x y y x y x y y y + + + + = + + + = = − + = − = = = 3 4 13 Equation of Tangent Line: ( - ) ( - ) ( -2) ( 1) y y m x x y x = = dy dx Find if sin 1 x x y ye + = d d d dx dx dx d d d d dx dx dx dx Implicit Differentiation ( sin ) ( ) (1) (sin ) sin ( ) ( ) ( ) 0 cos ' sin (1) ( ) ' 0 cos ' ' sin '( cos ) sin sin ' cos x x x x x x x x x x x x y ye x y y x y e e y x yy y y e e y x yy e y y ye y x y e y ye y ye y x y e + = + + + = + + + = + = − + = − = + (Doc #011p.36.01t)(Spring 2005/Fall1999)
Math1011 - Learning Strategies Center Various Prelims B2 dy dx Find if x 1 y + = dy 2 dx Find if ( 1) x y x + = + dy dx Find if sin x cos 1 y + =

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Math1011 Various Prelims B2 dy dx Find if x 1 y + = 1/2 1/2 d d d dx dx dx 1/2 1/2 1 1 2 2 1 1 2 x 2 y 1 -1 2 y 2 x -2 y - y 2 x x ( ) ( ) (1) Implicit Differentiation ' 0 ' 0 ' ' x y x y y y y y + = + = + = = = = dy 2 dx Find if ( 1) x y x + = + 2 d dx dy 1 y dx dy 1 1 y 1 dx dy 2 1 dx dy 2 2 1 dx ln ln( 1) ln ( 2)ln( +1) (ln ) (( 2)ln( +1)) ( 2) (ln( +1)) ln( +1) ( 2 ( 2) ln( +1) [ ln( +1)] ( 1) [ ln( +1)] x d dx d d dx dx x x x x x x ) y x y x x y x x x x x x x x y x x x + + + + + + + = + = + = + = + + + = + + = + = + + dy dx Find if sin x cos 1 y + = 1/2 1/2 d d d dx dx dx 1/2 1/2 1 1 2 2 1 1 2 x 2 y sin -cos 2 y 2 x 2 y cos 2 y y cos -cos 2 x sin sin 2 x x sin Implicit Differentiation (sin ) (cos ) (1) cos sin ' 0 cos sin ' 0 ' ' y x x x x y y y x y x x y y y x y y y y + = + − = = = = = = (Doc #011p.36.02t)( Fall2003)(Spring 2006)
Math1011 Spring 2007/Fall 2007 B3 A cubical block of ice melts, then refreezes, then melts again, and so on. The volume of the cube (in m³) at time t (in days) is V(t)=sin(2 π t)+2. Find the average rate of change in the Volume between the start of day 0 and 6 hours later.

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