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# Calc02_4 - 2.4 Rates of Change and Tangent Lines Devils...

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2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 1993

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The slope of a line is given by: y m x = x y The slope at (1,1) can be approximated by the slope of the secant through (4,16). 15 3 = 5 = We could get a better approximation if we move the point closer to (1,1). ie: (3,9) 8 2 = 4 = Even better would be the point (2,4). y x 4 1 2 1 - = - 3 1 = 3 = ( 29 2 f x x =
The slope of a line is given by: y m x = x y If we got really close to (1,1), say (1.1,1.21), the approximation would get better still .21 .1 = 2.1 = How far can we go? ( 29 2 f x x =

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( 29 1 f 1 1 h + ( 29 1 f h + h slope y x = ( 29 ( 29 1 1 f h f h + - = slope at ( 29 1,1 ( 29 2 0 1 1 lim h h h + - = 2 0 1 2 1 lim h h h h + + - = ( 29 0 2 lim h h h h + = 2 = The slope of the curve at the point is: ( 29 y f x = ( 29 ( 29 , P a f a ( 29 ( 29 0 lim h f a h f a m h + - =
The slope of the curve at the point is: ( 29 y f x = ( 29 ( 29 , P a f a ( 29 ( 29 0 lim h f a h f a m h + - = ( 29 ( 29 f a h f a h + - is called the difference quotient of f at a .

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