Table Of ContentAP CALCULUS BC
CHAPTER 8-10 NOTES
Alonzo and Tracy Mourning
Senior High
Mrs. Berkson
[email protected]
http://teachers.dadeschools.net/Lberkson
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
8.2
L’Hôpital’s Rule
Objec;ves:
• Find
limits
of
indeterminate
forms
using
L’Hôpital’s Rule.
x2−4
Consider: lim
x→2 x−2
If we try to evaluate this by direct substitution, we get:
Zero divided by zero can not be evaluated, and is an example of
indeterminate form.
→
f(x) x2−4
lim =lim
x→ag(x) x→2 x−2
L’Hôpital’s Rule:
f(x)
If l i m is indeterminate, then:
x→ag(x)
f(x) f′(x)
lim =lim
x→ag(x) x→ag′(x)
→
1
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
Example:
1−cosx
lim
x→0 x+x2
→
On the other hand, you can apply L’Hôpital’s rule as
many times as necessary as long as the fraction is still
indeterminate:
x
1+−x −1
lim 2
x→0 x2
→
L’Hôpital’s rule can be used to evaluate other
indeterminate 0
forms besides .
0
The following are also considered indeterminate:
∞
∞⋅0 ∞−∞ 1∞ 00 ∞0
∞
∞ 0
The first one, , can be evaluated just like .
∞ 0
The others must be changed to fractions first.
→
2
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
⎛ 1⎞
lim⎜xsin ⎟
x→∞⎝ x⎠
→
⎛ 1 1 ⎞
lim⎜ − ⎟
x→1⎝lnx x−1⎠
→
Indeterminate Forms: 1∞ 00 ∞0
Evaluating these forms requires a mathematical trick to change the
expression into a fraction.
limf(x)=⇒L = limf(x)=limelnf(x) eL
x→a x→a x→a
limx1/x
x→∞
p
3
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
8.3
Rela;ve
Rates
of
Growth
Objec;ves:
• Use
liKle-‐oh
and
big-‐oh
nota;on
in
determining,
inves;ga;on,
and
comparing
the
rates
of
growth
of
func;ons.
Definitions: Faster, Slower, Same-rate Growth as x→∞
Let f (x) and g(x) be positive for x sufficiently large.
1. f grows faster than g (and g grows slower than f )
as x → ∞ if
f(x) g(x)
lim =∞ or lim =0
x→∞g(x) x→∞ f(x)
2. f and g grow at the same rate as x → ∞ if
f(x)
lim =≠L 0
x→∞g(x)
→
According to this definition, y = 2 x does not grow faster
than y = x .
2x
6 lim =lim2=2
5 x→∞ x x→∞
4
3 Since this is a finite non-zero
2 limit, the functions grow at
1 the same rate!
-6-5-4-3-2-10 1 2 3 4 5 6
-1
-2 The book says that “f grows
y=x -3 faster than g” means that for
-4
-5 large x values, g is negligible
y=2x -6 compared to f.
“Grows faster” is not the same as “has a steeper slope”!
→
4
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
Which grows faster, e x or x 2 ?
ex This is indeterminate, so
lim
x→∞x2 we apply1 0L’Hôpital’s rule.
9
8
Still indet7erminate.
6
5
4
3
2
1
0 1 2 3 4 5 6 7 8 9 10
→
“Growing at the same rate” is transitive.
If f grows at the same rate as g as x → ∞ and g grows
as the same rate as h as x → ∞, then f grows at the
same rate as h as x → ∞.
In other words, if two functions grow at the same rate
as a third function, then the first two functions grow at
the same rate.
f g
lim =L lim =M
x→∞g x→∞h
f ⎛ f g⎞
lim ==⋅lim⎜ ⎟ LM
x→∞ h x→∞⎝g h⎠ →
Example 4:
Show that f ( x ) = x 2 + 5 and g ( x ) = − ( 2 x 1 ) 2 grow at
the same rate as x → ∞ .
→
5
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
Order and Oh-Notation
Definition f of Smaller Order than g
Let f and g be positive for x sufficiently large. Then f
is of smaller order than g as x → ∞ if
f(x)
lim =0
x→∞g(x)
We write f = o ( g ) and say “f is little-oh of g.”
Saying f = o ( g ) is another way to say that f grows
slower than g.
→
Order and Oh-Notation
Definition f of at Most the Order of g
Let f and g be positive for x sufficiently large. Then f
is of at most the order of g as x → ∞ if there is a
positive integer M for which
f(x)
≤M for x sufficiently large
g(x)
We write f = O ( g ) and say “f is big-oh of g.”
Saying f = O ( g ) is another way to say that f grows
no faster than g. p
8.4
Improper
Integrals
Objec;ves:
• Use
limits
to
evaluate
improper
integrals
6
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
Until now we have been finding integrals of continuous functions over
closed intervals.
Sometimes we can find integrals for functions where the function or the
limits are infinite. These are called improper integrals.
→
Example 1:
1 1+x
∫ dx
0 1−x
→
Example 2:
1dx
∫
0 x
We approach the limit from inside the interval.
→
7
Chapter
8-‐
L’Hopital’s
Rule,
Improper
Integrals,
and
Par;al
Frac;ons
Example 3:
3 dx
∫
0(x−1)23
→
Example 4:
∞dx
∫ P>0
1 xP
(P is a constant.)
p
8.4
Tests
for
Convergence
Objec;ves:
• Use
limits
to
evaluate
improper
integrals
• Use
the
direct
comparisons
test
and
the
limit
comparison
test
to
determine
the
convergence
or
divergence
of
improper
integrals
8
Description:AP CALCULUS BC. CHAPTER 8-10 NOTES. Alonzo and Tracy Mourning. Senior High. Mrs. Berkson.
[email protected].
