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The final exam for math 30, held on december 15, 2007. The exam covers various topics including integration, differential equations, and the logistics equation. Students are required to find volumes, average values, evaluate indefinite integrals, solve differential equations, and determine equilibrium points. The document also includes useful identities and information about the logistics equation.
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2
2 u x = x x
sin ( x )cos ( x ) dx
2 3
−
dx
x x 9
1
2 2
dx
x x
x
5 6
1
2
y x dt
dy sin
, )^2 2
( =
π y
dx
dy
Slope field for y’=x+y
y xy dt
dy
x xy dt
dx
( 1 cos( 2 )) 2
sin ( )
cos ( ) sin ( ) 1
sec ( ) tan ( ) 1
2
2 2
2 2
x x
x x
x x
tan( )sec( )
sec( )
sec ( )
tan( ) 2
d
d
d
d
0
1 exp( )
A kt
Pt
kPK P dt
dP