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17_Cal_Solution of Calculus_6e

# 17_Cal_Solution of Calculus_6e - f x:= Floor 2x Plot f x...

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f[x ] := Floor[ 2 x ]; Plot[ f[x], { x, 3 . 5, 3 . 5 } , AspectRatio > Automatic, PlotRange > { { − 3 . 5, 3 . 5 } , {− 4 . 5, 4 . 5 }} ]; Mathematica will draw vertical lines connecting points that it shouldn’t, making the graph look like treads and risers of a staircase, whereas only the treads are on the graph. C01S02.054: The function f is undefined at x = 1. The graph consists of the horizontal line y = 1 for x > 1 together with the horizontal line y = 1 for x < 1. There is a discontinuity at x = 1. C01S02.055: Given: f ( x ) = [[ x ]]. If n is an integer and n x < n + 1, then express x as x = n + (( x )) where (( x )) = x [[ x ]] is the fractional part of x . Then f ( x ) = n x = n [ n +(( x ))] = (( x )). So f ( x ) is the negative of the fractional part of x . So as x ranges from n up to (but not including) n + 1, f ( x ) begins at 0 and drops linearly down not quite to 1. That is, on the interval ( n, n + 1), the graph of f is the straight line segment connecting the two points ( n, 0) and ( n +1 , 1) with the first of these points included and the second excluded. There is a discontinuity at each integral value of
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