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Slope

The slope describes the steepnes of a curve in a certain point.

 


Explanation

A straight line through the point (a, 0) can be written as

m (x − a)

The slope m is then

The slope is also called the tangent. With it you can determine the rate of rise of a continuous function.

 


Example 1

Investigate the inflection points of the function f (x) = x4 − 6x3 + 12x2 − 8x + 1

f ′(x) = 4x3 − 18x2 + 24x − 8
f ′′(x) = 12x2 − 36x + 24
f ′′′(x) = 24x − 36

There are inflection points if f ′′(x) = 0 and f ′′′(x) ≠ 0

12x2 − 36x + 24 = 0     ⇒        x2 − 3x + 2 = 0     ⇒        x1 = 1           x2 = 2

f ′′′(1) = 24 − 36 = −12    ⇒    B1 (1, 0)
f ′′′(2) = 48 − 36 = 12      ⇒    B2 (2, 1)

The slope of an inflection point is m = f ′(xb)

B1 (1, 0)          ⇒          m1 = f ′(1) = 4 − 18 + 24 − 8 = 2      ascending
B2 (2, 1)          ⇒          m2 = f ′(2) = 32 − 72 + 48 − 8 = 0     horizontal

 


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