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# ch2extra - Using the definition f x h =-2 x h 3 =-2 x 2 h 3...

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Definition of the Derivative Using Average Rate (Page 129 - 133 and 160 in the book) a a+h f(a) Slope of the line = h f(a+h ) Average Rate of Change = f ( a+h ) – f ( a ) h f ( a+h ) – f ( a ) h

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a+h h a a+h h a a h Now, Watch what happens when: Point a is fixed and the size of the interval h shrinks a+h h a
As h shrinks and approaches zero (but not = 0), Slope of the Tangent line = f ( a+h ) – f ( a ) h h f(a+h ) f(a) a+h a Slope of the line = Average Rate of Change = f ( a+h ) – f ( a ) h the line becomes a Tangent Line As h approaches zero

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f(a) a lim: Limit, as h approaches zero f ( a+h ) – f ( a ) h = lim h 0 Slope of the Tangent line As h approaches zero, or: f ( a+h ) – f ( a ) h = The slope of the Tangent Line at a is the Derivative, f ' ( a ) f ( a+h ) – f ( a ) h lim h 0 f ' ( a ) =
Example: Use the definition of the derivative to obtain the following result: If f ( x ) = -2 x + 3, then f ' ( x ) = -2 Solution:

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Unformatted text preview: Using the definition f ( x + h ) = -2( x + h ) + 3 = (-2 x- 2 h + 3 ) = -2 f ( x+h ) – f ( x ) h f ' ( x ) = lim h = (-2 h ) h lim h = (-2 x- 2 h + 3 ) – (-2 x + 3) h lim h f ( x + h ) – f ( x ) h f ' ( x ) = lim h Example: Use the definition of the derivative to obtain the following result: If f ( x ) = x 2- 8 x + 9, then f ' ( x ) = 2 x- 8 f ( x + h ) = ( x + h ) 2- 8( x + h ) + 9 = ( x 2 + 2 xh + h 2- 8 x-8 h + 9 ) = 2 x - 8 f ( x+h ) – f ( x ) h f ' ( x ) = lim h = (2 x + h- 8) lim h = h (2 x + h- 8) h lim h = (2 xh + h 2- 8 h ) h lim h = ( x 2 + 2 xh + h 2- 8 x- 8 h + 9 ) – ( x 2- 8 x + 9) h lim h f ( x + h ) – f ( x ) h f ' ( x ) = lim h Solution: Using the definition...
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ch2extra - Using the definition f x h =-2 x h 3 =-2 x 2 h 3...

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