LECTURE NOTES: 4-3 HOW DERIVATIVES AFFECT THE SHAPE OFA GRAPH(PART 1)

MOTIVATING EXAMPLE: For each function graphed below, identify the regions of its domain where thefunction is increasing and where it is decreasing.

f(x) is increasing:

f(x) is decreasing:

f(x) is increasing:

f(x) is decreasing:

QUESTION 1: Using language a middle school kid could understand, how would you explain what itmeans to say a function is increasing or decreasing?

QUESTION 2: Draw a few sample tangent lines to each graph above. What is the relationship betweenthe slope of the tangent lines and whether the graph is increasing or decreasing?

Increasing/Decreasing Test

(a) If on an interval, then the function f(x) is increasing on this interval.

(b) If on an interval, then the function f(x) is decreasing on this interval.

1 4-3 Derivatives & Shape of Graph (part 1)

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If graph C or y- values ) go up ,

we say function Is increasing .

If graph ( or y- values ) go down

,then decreasing

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PRACTICE PROBLEM 1: Let g(x) = 3x4 � 4x3 � 12x2 + 5.

1. Use the Increasing/Decreasing Test to find the intervals where g(x) is increasing and decreasing.

2. Sketch the graph on your calculator to check that your answer above is correct.

3. What do you observe about the relationship between local maximums, local minimums and intervalsof increase and decrease? Make an explicit conjecture.

4. Go back and look at the examples at the top of page 1 and see if you need to amend your conjectureabove.

2 4-3 Derivatives & Shape of Graph (part 1)

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QUESTION 3: What is a critical number again?

The First Derivative Test: Suppose that c is a critical number of a continuous function f(x).

a) If at c, then f

has a local maximum at c.

b) If at c, then f

has a local minimum at c.

c) If at c, then f

has no local maximum or minimum at c.

Using the work from the previous PRACTICE PROBLEM 1 , fill in the blanks below for g(x) = 3x4 �4x3 � 12x2 + 5.

(a) is a local minimum of g(x) that occurs at

(b) is a local minimum of g(x) that occurs at

(c) is a local maximum of g(x) that occurs at

PRACTICE PROBLEM 2: Sketch the graph of a function h(x) satisfying all the properties below:

1. domain (�1,1)

2. h

0(x) > 0 on (�1, 0) [ (2,1)

3. h

0(x) < 0 on (0, 2)

4. h

0(0) is undefined, h0(2) = 0

3 4-3 Derivatives & Shape of Graph (part 1)

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f '4) does not change sign

y= 0 X= - 1

y= -27 X= 2

y=5 X=o

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MOTIVATING EXAMPLES: On the sample graphs below, sketch some rough tangent lines. Sketch mul-tiple tangents on each graph. Make rough approximations of the slopes of these tangents.

concave uppictures

concavedownpictures

QUESTION 4: How does the relationship between the tangent line and the graph to which it is tangentdiffer depending on whether the graph is concave up or concave down?

QUESTION 5: If the graphs above are of a function, say f(x), what can you say about its derivativef

0(x)?

QUESTION 6: Estimate the intervals where each function below is concave up and concave down:

f(x) is concave up:

f(x) is concave down:

f(x) is concave up:

f(x) is concave down:

4 4-3 Derivatives & Shape of Graph (part 1)

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DEFINITION: A point P on a curve y = f(x) is called an inflection point if f is continuous thereand the curve changes from concave upward to concave downward or vice versa at P .

QUESTION 6: Do the graphs on the previous page have any inflection points?

CONCAVITY TEST & INFLECTION POINTS: Let f(x) be a function defined on an interval I .

a) If (that is: ) for all x in I , thenthe graph of f is concave upward on I .

b) If (that is: ) for all x in I , thenthe graph of f is concave downward on I .

PRACTICE PROBLEM 3: Let f(x) = 2x3 � 3x2 � 12x. Find the intervals of concavity and the inflectionpoints.

5 4-3 Derivatives & Shape of Graph (part 1)

graph on left : 3 inflection points .marked in red .

graph on right :O inflection points .

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f "< o f '

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