L’Hospital’s Rule: f and g – differentiable, g(a) 0 near a (except possibly at a). If...
6
L’Hospital’s Rule: f and g – differentiable, g(a) 0 near a (except possibly at a). If (indeterminate form of type ) or (indeterminate form of type ) then (if this limit exists). 0 ) ( lim 0 ) ( lim x g and x f a x a x ) ( lim ) ( lim x g and x f a x a x 0 0 ) ( ) ( lim ) ( ) ( lim x g x f x g x f a x a x ? ) ( ) ( lim x g x f a x
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Transcript of L’Hospital’s Rule: f and g – differentiable, g(a) 0 near a (except possibly at a). If...
L’Hospital’s Rule:
f and g – differentiable, g(a) 0 near a (except possibly at a).
If (indeterminate form of type )
or (indeterminate form of type )
then (if this limit exists).
0)(lim0)(lim
xgandxfaxax
)(lim)(lim xgandxfaxax
0
0
)(
)(lim
)(
)(lim
xg
xf
xg
xfaxax
?)(
)(lim xg
xfax
Indeterminate Product
If (indeterminate form of type )
then use L’Hospital Rule for
)(lim0)(lim xgandxfaxax
0
.)(1
)(lim
)(1
)(lim
xf
xgor
xg
xfaxax
Indeterminate Difference
If (indeterminate form of type )
then try to convert into quotent and use L’Hospital Rule.
)(lim)(lim xgandxfaxax
?)()(lim
xgxfax
?)()(lim
xgxfax
Indeterminate Power
If (indeterminate form of type )
or (indeterminate form of type )
or (indeterminate form of type )
then use
0)(lim0)(lim
xgandxfaxax
00
?)(lim )(
xg
axxf
0)(lim)(lim
xgandxfaxax
)(lim1)(lim xgandxfaxax
0
1
).(ln)()(ln)( )()(ln)()( xfxgxforexf xgxfxgxg
Improper Integrals
Type I:
Infinite IntervalType II:
Discontinuous Integrand
.)()()(
)()().
;)(lim)()().
;)(lim)()().
a
a
a
a
b
tt
bb
t
t
at
a
t
a
dxxfdxxfdxxf
convergentdxxfanddxxfaIfc
dxxfdxxfbtexistsdxxfIfb
dxxfdxxfatexistsdxxfIfa
Type I Improper Integral:
If limit exists then this integral is con
vergent,
otherwise – divergen
t.
Type II Improper Integral:
a). If f – continuous on [a,b) and discontinuous at b
b). If f – continuous on (a,b] and discontinuous at a
c). If f – discontinuous at c(a,b) and
;)(lim)(
t
abt
b
a
dxxfdxxf
;)(lim)(
b
tat
b
a
dxxfdxxf
.)()()(
)()(
b
c
c
a
b
a
b
c
c
a
dxxfdxxfdxxf
convergentdxxfanddxxf
If limit exists then this integral is con
vergent,
otherwise – divergen
t.
Comparison Th for Improper Integrals:f and g – continuous, f(x)g(x) 0 for xa.
.)()().
;)()().
divergentdxxfdivergentdxxgIfb
convergentdxxgconvergentdxxfIfa
aa
aa
Differential equation –
Equation that contains an unknown function and some of its derivatives.
Order of differential equation = order of highest derivative in equations.
f(x) – solution of differential equation, if it satisfies the equation for all x in some interval.
Solve differential equation = find ALL possible solutions.
Condition of the form y(t0)=y0 – initial condition.
Initial-value Problem (IVP) – find solution that satisfies the given initial condition.
Separable equation –
1st order differential equation in which expression for dy/dx can be factored as a function of x times a function of y:
)()( yfxgdx
dy
dxxgdyyh
yfifyf
yhdxxgdyyhyfxgdx
dy
)()(
0)()(
1)()()()()(
Mixing problem:
A tank contains 20 kg of salt dissolved in 5000L of water. Brine that contains 0.03 kg of saltper liter of water enters the tank at a rate of 25L/min. The solution is kept throughly mixedand drains from the tank at the same rate. How much salt remains in the tank after half an hour?
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