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Created by
Jason L. Bradbury
State Standard
– 15.0b Students use the fundamental theorem of calculus to interpret integrals as antidervivatives.
– 14.0 Students apply the definition of the integral to model problems of physics, economics, and so forth obtaining results in terms of integrals.
Objective – To be able to use the 2nd derivative test to find concavity and points of inflection.
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Example 1
Find the general solution fo the differential equation 2y
2y x C
2y x
To begin, you need to find a function whose derivative is 2.
Now we can apply the Theorem
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Differentiation Formula Integration Formula
0dx C 0dc
dx
kdx kx C dkx k
dx
1
1
nxC
n
1n nd
x nxdx
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
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Differentiation Formula Integration Formula
cos sinxdx x C
sin cosxdx x C 2sec tanxdx x C
cos sind
x xdx
sin cosd
x xdx
sec sec tand
x x xdx
2tan secd
x xdx
2cot cscd
x xdx
csc csc cotd
x x xdx
sec tan secx xdx x C
csc cot cscx xdx x C 2csc cotxdx x C
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2x dx2 1
3
xc
3dx0 1
31
xc
3x c 3
3
xc
2
1dx
x2x dx
2 1
1
xc
1c
x
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23 2 2x x dx 23 2 2x dx xdx dx
2 1 1 1 0 1
3 2 23 2 1
x x xc
3 2 2x x x c
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3cos xdx 3sin x c 25sec xdx 5 tan x c
4sin csc cotx x x dx4sin csc cotxdx x xdx
4cos cscx x c
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2dxx
1
22x dx
1
12
212
xc
1
24x c
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22 1t dt 4 22 1t t dt 4 22 1t dt t dt dt 4 1 2 1 0 1
2 15 3 1
t t tc
5 31 2
5 3t t t c
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3
2
3xdx
x
3
2 2
3xdx
x x
23x x dx 23xdx x dx 1 1 2 1
32 1
x xc
21 3
2x c
x
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xe dxxe c
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Here are some examples from the homework
7x dx7 1
8
xc
2)
8
8
xc
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Here are some examples from the homework
3 2x dx2
13
53
xc
16)
5
33
5
xc
2
3x dx3 23
5
x xc
3 53
5
xc
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Here are some examples from the homework
5xe dx
5u due
28)
1
5ue du
5u x
51
5xe c
1
5ue c
5du dx5
dudx