l`Hospital`s rule, Differential Equations and Integration

Math 104: l’Hospital’s rule, Differential Equations and
Integration
Ryan Blair
University of Pennsylvania
Tuesday January 22, 2013
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Integration
January 22, 2013
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Outline
1
l’Hospital’s rule
2
Differential Equations
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Integration
January 22, 2013
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l’Hospital’s rule
l’Hospital’s rule
Last time we saw how to find certain
0
0
limits using Taylor Series.
Theorem
If
limx→a f (x) = 0 = limx→a g (x),
Then
Ryan Blair (U Penn)
f (x)
f ′ (x)
limx→a
= limx→a ′ .
g (x)
g (x)
Math 104:l’Hospital’s rule, Differential Equations andTuesday
Integration
January 22, 2013
3/8
l’Hospital’s rule
l’Hospital’s rule
Last time we saw how to find certain
0
0
limits using Taylor Series.
Theorem
If
limx→a f (x) = 0 = limx→a g (x),
Then
f (x)
f ′ (x)
limx→a
= limx→a ′ .
g (x)
g (x)
Exercise: Find the following limit using l’Hospital’s
limx→0
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ex − 1 − x
x2
Math 104:l’Hospital’s rule, Differential Equations andTuesday
Integration
January 22, 2013
3/8
l’Hospital’s rule
l’Hospital’s rule
Last time we saw how to find certain
0
0
limits using Taylor Series.
Theorem
If
limx→a f (x) = 0 = limx→a g (x),
Then
f (x)
f ′ (x)
limx→a
= limx→a ′ .
g (x)
g (x)
Exercise: Find the following limit using l’Hospital’s
limx→0
ex − 1 − x
x2
Exercise: Use Taylor series to prove l’Hospital’s theorem supposing
f ′ (a)
is well defined
g ′ (a)
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Integration
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Differential Equations
Differential Equations
Definition
An nth order Ordinary Differential Equation (O.D.E.) on y (x) is an
algebraic equation including y and its derivatives
dy d 2 y
d ny
, 2 , ..., n ) = 0
dx dx
dx
A solution is any function that satisfies the above equation.
F (x, y (x),
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Integration
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Differential Equations
Differential Equations
Definition
An nth order Ordinary Differential Equation (O.D.E.) on y (x) is an
algebraic equation including y and its derivatives
dy d 2 y
d ny
, 2 , ..., n ) = 0
dx dx
dx
A solution is any function that satisfies the above equation.
F (x, y (x),
Example: An object near the surface of the earth acted on by gravity
= v ′ (t) = −g
has velocity v (t) given by dv
dt
Solve for v (t).
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Integration
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Differential Equations
Initial value problem
Definition
An initial value problem is an O.D.E. together with an initial
condition given by y (0) = c for some constant c.
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Integration
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Differential Equations
Initial value problem
Definition
An initial value problem is an O.D.E. together with an initial
condition given by y (0) = c for some constant c.
Example: Solve the following IVP
dy
= cos(2x) and y (0) = −1.
dx
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Math 104:l’Hospital’s rule, Differential Equations andTuesday
Integration
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Differential Equations
Summary of Integration Techniques
To solve O.D.E.s we need to be able to integrate.
1
2
3
4
5
u substitution
integration by parts
trigonometric substitutions
partial fractions
completing the square
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Integration
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Differential Equations
U-Substitution
Z
′
f (g (x))g (x)dx =
Z
f (u)du
The main difficulty is determining u = g (x).
.
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Integration
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Differential Equations
U-Substitution
Z
′
f (g (x))g (x)dx =
Z
f (u)du
The main difficulty is determining u = g (x).
Exercise Find
R
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3
x 2 e x dx.
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Integration
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Differential Equations
U-Substitution
Z
′
f (g (x))g (x)dx =
Z
f (u)du
The main difficulty is determining u = g (x).
Exercise Find
Exercise Find
R
R
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3
x 2 e x dx.
(1+ln(x))3
dx.
x
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Integration
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Differential Equations
Integration by Parts
Z
′
u(x)v (x)dx = u(x)v (x) −
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Z
u ′ (x)v (x)dx
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Integration
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Differential Equations
Integration by Parts
Z
′
u(x)v (x)dx = u(x)v (x) −
Z
u ′ (x)v (x)dx
Example: Derive the above formula from the product rule for
derivatives and the fundamental theorem of calculus.
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Integration
January 22, 2013
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Differential Equations
Integration by Parts
Z
′
u(x)v (x)dx = u(x)v (x) −
Z
u ′ (x)v (x)dx
Example: Derive the above formula from the product rule for
derivatives and the fundamental theorem of calculus.
Example: Find
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R
xe x dx.
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Integration
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