SECOND DERIVATIVE TEST Section 3 4 Calculus APDual

SECOND DERIVATIVE TEST Section 3. 4 Calculus AP/Dual, Revised © 2015 viet. dang@humble. k 12. tx. us 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 1

FILL IN THE BLANK REVIEW 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 2

FILL IN THE BLANK TABLE INCREASING CRITICAL POINTS, POSSIBLE EXTREMA DECREASING CONCAVE UP ABOVE X– AXIS ZERO or UND BELOW X– AXIS INCREASING (increasing) CONCAVE DOWN DECREASING (decreasing) ABOVE X– AXIS (increasing) BELOW the graph (Relative Min) Tangent Line 1/13/2022 6: 41 AM POSSIBLE POINT(S) OF INFLECTION 3. 4 - Second Derivative Test ZERO or UND BELOW X– AXIS (decreasing) ABOVE the graph (Relative Max) 3

REVIEW EXAMPLE 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 4

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REVIEW EXAMPLE 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 6

REVIEW EXAMPLE 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 7

F’ VS F’’ CRITICAL POINTS ABSOLUTE/GLOBAL EXTREMA RELATIVE/LOCAL MAX OR MIN 1/13/2022 6: 41 AM CONCAVITY POINT OF INFLECTION PROCESS F ’’ = 0 OR DNE 3. 4 - Second Derivative Test 8

SECOND DERIVATIVE TEST 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 9

STEPS A. Identify all critical points using the first derivative. B. Determine the second derivative C. Plug in the critical points into the second derivative 1. If the critical point is positive, there is a relative max 2. If the critical point is negative, there is a relative min 3. If the critical point is zero, test is inconclusive and test must revert back to the first derivative test 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 10

EXAMPLE 1 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 11

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EXAMPLE 2 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 15

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EXAMPLE 2 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 17

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YOUR TURN 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 19

EXAMPLE 3 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 20

EXAMPLE 4 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 21

EXAMPLE 4 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 22

EXAMPLE 4 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 23

TO RECAP A. The First Derivative tells us: 1. Extrema 2. Relative Maximum 3. Relative Minimum B. The Second Derivative tells us: 1. Concavity 2. Points of Inflection C. The Second Derivative TEST tells us: 1. Extrema 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 24

AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR) 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 25

ASSIGNMENT Page 192 31 -40 all & Justify all responses 1/13/2022 6: 41 AM 3. 4 - Second Derivative Test 26