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1 Concavity and the Second Derivative Test Section 3.4

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2 Concavity Theorem 3.7- Test for Concavity Let f be a function whose second derivative exists on an open interval I. 1.If f ´´(x) > 0 for all x in I, then the graph of f is concave upward in I. (“Holds water”) 2.If f ´´(x) < 0 for all x in I, then the graph of f is concave downward in I. (“Spills water”)

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3 Concavity

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5 Points of Inflection Concavity changes at points of inflection. Examples: If is a point of inflection of the graph of f, then either _____________ or ___________________________.

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6 Concavity To determine the intervals in which the graph of f is concave upward or concave downward, do the following. 1.Find the second derivative of f 2.Locate the x-values at which f ’’(x) = 0 or f ” does not exist. 3.On a number line, mark down these values to create our test intervals. 4.Test the sign of f ”(x) in each of the test intervals to determine where the second derivative is positive (concave up) or negative (concave down). 5.The points of inflection are those x-values at which f ”(x) changed sign.

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7 Example 1 Find the points of inflection and discuss the concavity of the graph of the function.

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8 Example 1 (continued)

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9 Example 2 Find the points of inflection and discuss the concavity of the graph of the function.

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10 Example 2 (continued)

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11 Second Derivative The second derivative can be used to identify relative minima and relative maxima as well.

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12 Second Derivative Test Theorem 3.9- Second Derivative Test Let f be a function such that f ´(c) = 0 and the second derivative of f exists on an open interval containing c. 1.If f ´´(x) > 0, then f(c) is a relative minimum. 2.If f ´´(x) < 0, then f(c) is a relative maximum. 3.If f´´(x) = 0, then the test fails. In this case, use the first derivative test.

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13 Example 3 Find all of the relative extrema. Use the second derivative test, where applicable.

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14 Example 3 (continued)

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15 Example 4 Find all of the relative extrema. Use the second derivative test, where applicable.

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16 Example 4 (continued)

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17 (Almost) Full Curve Sketch Let’s put together what we have learned so far about curve sketching. We will consider the following features to help us to graph the given function by hand. 1.Domain 2.First derivative 3.Second derivative. 4.Intervals for which graph is increasing/deceasing/level; intervals for which graph is concave upward/downward 5.Coordinates of relative extrema, points of inflection, intercepts.

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18 (Almost) Full Curve Sketch- 7 parts Example: 1. Domain: 2. 1 st Derivative:

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nd Derivative: (Almost) Full Curve Sketch- 7 parts

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20 4. Limits of Special Interest: (Almost) Full Curve Sketch- 7 parts We will look at these in the next section.

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21 5. Summary: Draw domain Look at 1 st Derivative Look at 2 nd Derivative (Almost) Full Curve Sketch- 7 parts

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22 6. Points: Relative Extrema: Points of Inflection: Intercepts: (Almost) Full Curve Sketch- 7 parts

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23 7. Graph: 8. Range, etc: Range: ___________ Abs. Max ________ Abs. Min ________ Increasing on ________________ Decreasing on ________________ (Almost) Full Curve Sketch- 7 parts Symmetry ____________________

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24 Homework Section 3.4 page 195 #1-7 odd, odd, odd, 37, 39, 53

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