ap calculus ab
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AP Calculus AB. Day 6 Section 3.4. Second Derivative Test. If c is a critical number of f’(x) and…. (c, f(c)) is a relative min. If f’’(c) > 0 then ______________________________________________. (c, f(c)) is a relative max. - PowerPoint PPT PresentationTRANSCRIPT
04/19/23 Perkins
AP Calculus AB
Day 6Section 3.4
Second Derivative Test
If f’’(c) > 0 then ______________________________________________
(c, f(c)) is a relative min
If c is a critical number of f’(x) and…
If f’’(c) < 0 then _______________________________________________
If f’’(c) = 0 or is undefined then __________________________________
(c, f(c)) is a relative max
the test fails (use 1st Derivative test)
1. Find the extrema of 2sin cos2 on 0,2f x x x
' 2cos 2sin2f x x x Crit numbers: 0 2cos 2sin2x x
'' 2sin 4cos2f x x x
0 2cos 2 2sin cosx x x 0 2cos 1 2sinx x
0 2cos 0 1 2sinx x 120 cos sinx x
3 52 2 6 6, ,x x
2'' 2f
32'' 6f
56'' 3f
6'' 3f 326f
2 1f
32 3f
25 36f
rel min at
rel max at
rel min at
rel max at
36 2,
2 ,1
5 36 2,
32 , 3
2nd Derivative Test
2. Sketch 22sin on 0,4xf x
2' cos xf x Crit numbers:
,3 ,5 ,7x
12'' 3f
12''f 2f
23f rel min at
rel max at ,2
3 , 2
2nd Derivative Test
20 cos x3
2 2 2,x 5 72 2,
12 2'' sin xf x Crit numbers: 20 sin x
2 ,2 ,3 ,4x 2 ,4 ,6 ,8x Intervals: 0, 2 ,4
Test values: 3
Inf pt
2x
2 0f
2 ,0
f ’’(test pt) f(x) down up
2
-2
2
2sin xf x
4
Perkins
AP Calculus AB
Day 6Section 3.4
Second Derivative Test
If f’’(c) > 0 then ______________________________________________
If c is a critical number of f’(x) and…
If f’’(c) < 0 then _______________________________________________
If f’’(c) = 0 or is undefined then __________________________________
1. Find the extrema of 2sin cos2 on 0,2f x x x
2. Sketch 22sin on 0,4xf x
2sin xf x
2
-2
24