Math1011_Week09 - Math1011 Section 3.7 Consider the curve...

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Math1011 Section 3.7 10/18/2010 2 2 Consider the curve x y - 6y + 2 = 0 Find the equation of the tangent line to the curve at point (2,1) 2 3 Given 5 tan r e θ θ θ = + + dr d Find θ d dr Find θ
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Math1011 Section 3.7 1 2 2 Consider the curve x y - 6y + 2 = 0 Find the equation of the tangent line to the curve at point (2,1) 2 2 2 2 2 2 d d d d dx dx dx dx 2 2 2 2 2 2 2 2 2 2 2 2 6 2(2)(1) 4 4 8 6 2 (2) 2(1) 6 Implicit Differentiation (x y ) (6 ) (2) (0) ( ) ( ) 6 ' 0 0 2 ' 2 6 ' 0 2 ' 6 ' 2 '( 2 6) 2 ' at point (2, 1), y'= 2 d d dx dx xy x y y x y y x y x yy y x y x yy y y x y x y xy y y y + = + + = + = = − = − = = = = − 1 1 ( ) 1 2( 2) equation of the line tangent to the curve at (2, 1) is 1 2( 2) m x x y x y x = = − = − 2 3 Given 5 tan r e θ θ θ = + + dr d Find θ d dr Find θ 2 3 2 3 2 3 5 tan 10 sec 3 10 sec 3 d d d d d d d d dr d dr d r e e e θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ = + + = + + = + + 2 3 2 3 2 3 2 3 1 10 sec 3 5 tan 1 10 sec 3 1 (10 sec 3 ) d d d d dr dr dr dr d d d dr dr dr d dr d dr e r e e e θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ + + = + + = + + = + + = (Doc #1011w.37.01)
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Math1011 Section 3.8 2 -1 2 Find (f )'(3) if f(x) = x 8+x 2 1 d 1 dx 1 Show that (tan ) x x + =
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Math1011 Section 3.8 2 -1 2 Find (f )'(3) if f(x) = x 8+x 1 1 1 '( ( )) -1 2 1/2 2 1 2 2 1/2 2 10 1 1 2 3 3 -1 3 1 10 10/3 ( )'( ) (3) 1 and '( ) ( )(8 ) (2 ) 8 '(1) (1)( )(8 (1) ) (2)(1) 8 (1) 3 (f )'(3) f f x f x f f x x x x x f = = = + + + = + + + = + = = = -1 Since (1) 3 implies that (3) 1 f f = = 2 1 d 1 dx 1 Show that (tan ) x x + = 1 2 1 2 2 2 2 2 1 1 '( ( )) 1 1 sec (tan ( )) 1 1 sec ( ) 1 1 ( 1) /1 1 1 1 ( ) tan '( ) sec ( )'( ) (tan ) (tan ) (tan ) (tan ) f f x d dx x d dx d dx x d dx x f x x f x x f x x x x x θ + + = = = = = = = θ x²+1 x 1 (Doc #1011w.38.01)(Adams 3.1.30)
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Math1011 Section 3.8 3 Differentiate the functions 2 ( ) ln( sin ) F t t t = + 2 1 2 3 Find f'(t) if ( ) ln t t f t + + =
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Math1011 Section 3.8 3 Differentiate the functions 2 ( ) ln( sin ) F t t t = + 2 2 2 2 d 1 dt 2 1 sin 1 sin 2 cos sin '( ) ln( sin ) Note: (ln ) '( ) ( sin ) '( ) (2 cos ) '( ) d t dt d dt t t t t t t t t F t t t t F t t t F t t t F t + + + + = + = + = + = = 2 1 2 3 Find f'(t) if ( ) ln t t f t + + = 2 1 2 3 1/2 2 1 2 3 1 2 1 2 1 1 1 2 2 1 2 3 1 1 1 2 2 1 2 3 2 1 1 1 1 2 2 1 2 3 2 1 2 3 ( ) ln ( ) (ln( ) ) ( ) (ln(2 1) ln(2 3)) '( ) (ln(2 1) ln(2 3)) '( ) [ (2 1) (2 3)] '( ) [ (2) (2)] '( ) [ ] t t t t d dt d d t t dt dt t t t t t t f t f t f t t t f t t t f t t t f t f t + + + + + + + + + + + + = = = + + = + + = + = = = + (Doc #1011w.38.02)
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Math1011 Section 3.8 4 sin Find the derivative of x y x = (x+2) Find the derivative of y=(x+1)
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