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week8 - Higher order derivatives Given y = f(x we have...

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Higher order derivatives Given y = f ( x ) , we have: first derivative: dy dx := f ( x ) second derivative: d 2 y dx 2 := d dx dy dx = f ( x ) = ( f ( x )) third derivative: d 3 y dx 2 := d dx d 2 y dx 2 = d dx d dx dy dx = = f ( x ) = ( f ( x )) = ( f ( x )) and so forth... Example (Ex. 16, Sect. 12.2) Find f ( x ) for f ( x ) = 2 x 5 Solution first derivative: f ( x ) = d dx (2 x 5 ) = 2 · 5 x 4 = 10 x 4 second derivative: f ( x ) = d 2 dx 2 ( 2 x 5 ) := d dx d dx (2 x 5 ) = d dx (10 x 4 ) = 10 · 4 x 3 = 40 x 3 third derivative: f ( x ) = d 3 dx 3 ( 2 x 5 ) = d dx d 2 dx 2 ( 2 x 5 ) = d dx (40 x 3 ) = 40 · 3 x 2 = 120 x 2
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Relative Extrema Definition A function f : I R is increasing if for inputs x 1 and x 2 such that x 1 < x 2 , the outputs have f ( x 1 ) and f ( x 2 ) have the same ordering f ( x 1 ) < f ( x 2 ) . If y = f ( x ) , we can write y = f ( x ) is increasing if for x 1 < x 2 we have y 1 < y 2 . If f is increasing, then when we look at the graph of f from left to right, it is going up-hill. Definition Analogous we have y = f ( x ) is decreasing if for x 1 < x 2 we have y 1 > y 2 , that is, when we look at the graph of f from left to right, it is going down-hill. Definition A function has a relative maximum (like a peak) at a if f ( x ) f ( a ) for all x in an interval containing a. Definition A function has a relative minimum at a if f ( x ) f ( a ) for all x in an interval containing a.
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