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.
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
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
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
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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