L ectur e 1 3| 1 Curvature and Inflection points
Definition
If the graph of is lying above its tangent line at every point in an interval , then it is called concave upward (CU) on .
L ectur e 1 3| 2
Definition
If the graph of is lying below its tangent line at every point in an interval , then it is called concave downward (CD) on .
L ectur e 1 3| 3
Definition
A point where the graph of changes its direction of concavity is called an
inflection point.
L ectur e 1 3| 4
Concavity test
(a) If for all in an interval , then the graph of is CU on .
(b) If for all in an interval , then the graph of is CD on .
L ectur e 1 3| 5 EX For each of the following functions,
find the intervals where the function is CU, where the function is CD, and the inflection points.
1.
2.
L ectur e 1 3| 6
Second Derivative Test
1. If and , then is a local min.
2. If and , then is a local max.
L ectur e 1 3| 7 EX For the function ,
find all the local extreme values of
L ectur e 1 3| 8 Curve Sketching
(a) Domain
(b) -interception points and - interception points
(c) Vertical and horizontal asymptotes (d) Intervals of increase/decrease
(e) Local max and local min
(f) Intervals of concavity and inflection points.
L ectur e 1 3| 9
I, CU I, CD
D, CU
D, CD
L ectur e 1 3| 10 EX Sketch the graph of .
L ectur e 1 3| 11 EX Sketch the graph of .
L ectur e 1 3| 12 EX Sketch the graph of
.
L ectur e 1 3| 13 EX Sketch the curve .
