1. 1 2 Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
f ’(x) > 0 for all x in I 2
f(x) increasing
Increasing\Decreasing
f ’ (x) < 0 for all x in I
f(x) decreasing

2. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

3. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

4. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

5. _ _ + + 1 2 1 2 First Derivative Test: First Derivative Test:
How do you know that: local maximum or local minimum
First Derivative Test: 1 _ + 2
First Derivative Test: 1 _ + 2

6. F081

7. F083

8. F102

9. 1 2 Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Concavity
f ’’(x) > 0 for all x in I 2
f(x) concave Up
Concavity
f ’’(x) < 0 for all x in I
f(x) concave Down

10. 1 2 Example: Discuss the curve with respect to concavity,
Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 1
f ’’(x) > 0 for all x in I 2
f(x) concave Up
Concavity
f ’’(x) < 0 for all x in I
f(x) concave Down
Example:
Discuss the curve
with respect to concavity,

11. Concavity Test 1
f ’’(x) > 0 for all x in I
f(x) concave Up

12. Concavity Test 1
f ’’(x) < 0 for all x in I
f(x) concave Down

13. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

14. _ _ + + 1 2 Inflection point:
Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
Inflection point: 1 _ + _ 2 +

15. Find all inflection points of
Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
Example:
Find all inflection points of

16. F091

17. 1 2 1 2 second Derivative Test: second Derivative Test:
How do you know that: local maximum or local minimum
second Derivative Test: 1 2
second Derivative Test: 1 2

18. F081

20. F091

21. F091

22. F083