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

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

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

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

_ _ + + 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

F081

F083

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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

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,

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

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

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

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

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

F091

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

F081

F091

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F083