ACTION Sketching graphs of fi rst and second derivatives
OBJECTIVE To sketch the graphs of the fi rst and second derivatives of functions
Local minimum (L Min)
As you walk from A to B and then to C around the valley with equation y = x2 – 4x + 8, the slope m of the tangent is increasing as x increases. The slope changes from negative values from A to B, to 0 at B, and then to positive values from B to C.
y m = −6 A D m − m = −2 B m = 0 m = +2 x The slope of the tangent to the curve at every point on this curve is increasing.
The actual values for the slopes are obtained by differentiation: y = x 2 − 4x + 8 dy
___ dx = 2x − 4 = m
m
2 4 6
–1 –2
–4 –6
(5, 6) = (x2, m2)
___ dx =
dm 0 12 3 4 5 (1, –2) = (x1, m1) x
_______ x2 – x1
m2 – m1
= 6 + 2 _____
5 – 1 = + 2
The slope m is increasing for all values of x ⇒ dm ___
___ dx =
dm d 2 y dx > 0 for all x ∈ R.
___ dx 2 > 0 for all points on this curve.
There are many points on this curve but there is only one point at which the curve fl attens out. This point is the bottom of the valley ( point B). It is known as a local minimum point of the curve. At B, the slope of the tangent is 0 as the curve fl attens out. At a local minimum point of a curve, two conditions hold:
1. 2.
___ dx = 0 at the point [slope = 0]
dy
___ dx 2 > 0 at the point [slope m is increasing from m(–) to m(+)]
