Post on 21-Dec-2015
transcript
L’HospitalL’HospitalAnother useful technique for computing limits
is L'Hospital's rule:
Basic version:
0)(lim)(lim
xgxfaxax
)(lim)(lim xgxfaxax
)('
)('lim
)(
)(lim
xg
xf
xg
xfaxax
If , then
provided the latter exists.
This also applies if
Fancy L’HospitalFancy L’Hospital
s.often work rule sHospital'L' applying
then and logarithm thegthen takin
,1or or 0 like lookslimit theIf 00
You can use “Basic L’Hospital” for #4, 6, 8 and 10 of the top ten limit list. But for limits like #9, you need...
3. (A) If 0 < x < 1 then 0lim
n
nx
(B) If x > 1, then
n
nxlim
4. and1sin
lim0
x
xx
0cos1
lim0
x
xx
0 e lim
x
x5. and
e lim x
x
6-106-10
6. For any value of n,
and for any positive value of n,
7.
1sin lim
0
xx
does not exist!
0lim x
n
x e
x
0ln
lim nx x
x
20
cos1lim
x
xx
564
32lim
2
2
xx
xxx
xxxx
1lim 2
Here are three more:Here are three more:
A challenge:
How about this one?How about this one?
2/1
0x
sinlim
x
x
x
A.
B. 0
C. -
D. 1
E.
F.
G.
H.
2e
2
6/1e
4/1e
Last one Last one (for now)...(for now)...
A.
B. 1/2
C.
D. 3
E. F. undefined
G.
H. 22e
2e2e
22 e
23e
xtx
xdtexe
0
22
lim
Improper Improper integralsintegrals
These are a special kind of limit. An improper integral is one where either the interval of integration is infinite, or else it includes a singularity of the function being integrated.
Examples of the second kind:Examples of the second kind:
32
4
tan and 11
0
dxxdxx
The second of these is subtle because the singularity of tan x occurs in the interior of the interval of integration rather than at one of the endpoints.
Same methodSame methodNo matter which kind of improper integral (or combination of improper integrals) we are faced with, the method of dealing with them is the same:
b
x
b
x dxedxe00
lim as thingsame themeans
Calculate the limit!Calculate the limit!What is the value of this limit
(and hence, of the improper integral )?
A. 0
B. 1
C.
D.
E.
ebe
dxe x
0
Another improper integralAnother improper integral
?1
1 of value theisWhat
.arctan1
1 that Recall
-2
2
dxx
Cxdxx
A. 0
B.
C.
4
2
D.
E.
Area between the x-Area between the x-axis and the graphaxis and the graph
The integral you just worked represents all of the area
between the x-axis and the graph of 211x
The other type...The other type...For improper integrals of the other type, we make the same kind of limit definition:
.1
lim be todefined is 1 1
0
1
0
dxx
dxx a
a
Another example:Another example:What is the value of this limit, in
other words, what is
A. 0
B. 1
C. 2
D.
E.
2
?11
0
dxx
A divergent improper integralA divergent improper integral
It is possible that the limit used to define the improper integral fails to exist or is infinite. In this case, the improper integral is said to diverge . If the limit does exist, then the improper integral converges. For example:
1
00
ln1ln lim1
adxx a
so this improper integral diverges.
Sometimes it is possible...Sometimes it is possible...
to show that an improper integral converges without actually evaluating it:
So the limit of the first integral must be finite as b goes to infinity, because it increases as b does but is bounded above (by 1/3).
.1 allfor 3
11
7
1
thathave we,0 allfor 1
7
1 Since
331
1 144
44
bb
dxx
dxxx
xxxx
b b
A puzzling example...A puzzling example...Consider the surface obtained by rotating the graph
of y = 1/x for x > 1 around the x-axis:
Let’s calculate the volume contained inside the surface:
units. cubic 1 lim 1
1
21
bbx dxV
What about the surface area?What about the surface area?This is equal to...
1
41
2 11
2)('1)(2 dx
xxdxxfxfSA
This last integral is difficult (impossible) to evaluate directly, but it is easy to see that its integrand is bigger than that of the divergent integral
Therefore it, too is divergent, so the surface has infinite surface area.
This surface is sometimes called "Gabriel's horn" -- it is a surface that can be "filled with water" but not "painted".
dxx
1
2
SequencesSequences
The lists of numbers you generate using a numerical method like Newton's method to get better and better approximations to the root of an equation are examples of (mathematical) sequences .
Sequences are infinite lists of numbers, Sometimes it is useful to think of them as functions from the positive integers into the reals, in other words,
,...,, 321 aaa
forth. so and ,a(2) ,a(1) 21 aa
The feeling we have about numerical methods like the bisection method, is that if we kept doing it more and more times, we would get numbers that are closer and closer to the actual root of the equation. In other words
where r is the root.
Sequences for which exists
and is finite are called convergent, other sequences are called divergent
Convergent and DivergentConvergent and Divergent
nn
a lim
rann
lim
For example...For example...
The sequence
1, 1/2, 1/4, 1/8, 1/16, .... , 1/2 , ... is convergent (and converges to zero, since
), whereas:
the sequence 1, 4, 9, 16, .…n , ... is divergent.
2
n
0 lim21
n
n
PracticePractice
The sequence
...,2
1,......,
5
4,
4
3,
3
2
n
n
A. Converges to 0
B. Converges to 1
C. Converges to n
D. Converges to ln 2
E. Diverges
Another...Another...
The sequence
...,2
1)1(,......,
5
4,
4
3,
3
2
n
nn
A. Converges to 0
B. Converges to 1
C. Converges to -1
D. Converges to ln 2
E. Diverges
A powerful existence theoremA powerful existence theoremIt is sometimes possible to assert that a sequence is convergent even if we can't find the limit right away. We do this by using the least upper bound property of the real numbers:
If a sequence has the property that a <a <a < .... is called a "monotonically increasing" sequence. For such a sequence, either the sequence is bounded (all the terms are less than some fixed number) or else it increases without bound to infinity. The latter case is divergent, and the former must converge to the least upper bound of the set of numbers {a , a , ... } . So if we find some upper bound, we are guaranteed convergence, even if we can't find the least upper bound.
1 2 3
1 2
Consider the sequence...Consider the sequence...
222,22 ,2
To get each term from the previous one, you add 2 and then take the square root.
It is clear that this is a monotonically increasing sequence. It is convergent because all the terms are less than 2. To see this, note that if x>2, then
So our terms can't be greater than 2, since adding 2 and taking the square root makes our terms bigger, not smaller.
Therefore, the sequence has a limit, by the theorem.
etc.
.2 so and ,222 xxxxx
Newton’s MethodNewton’s MethodA better way of generating a sequence of numbers that are (usually) better and better solutions of an equation is called Newton's method.
In it, you improve a guess at the root by calculating the place where the tangent line drawn to the graph of f(x) at the guess intersects the x-axis. Since the tangent line to the graph of y = f(x) at x = a is y = f(a) + f '(a) (x-a), and this line hits the x-axis when y=0, we solve for x in the equation f(a) + f '(a)(x-a) = 0 and get x = a - f(a)/f '(a).
)('
)(
old
oldoldnew xf
xfxx
Let’s try itLet’s try it
on the same function we used before,
with the guess that the root x1 = 2. Then the next guess is
This is 1.8. Let's try it again. A calculator helps:
22)( 3 xxxf
5
9
10
22
)('
)(
1
112
xf
xfxx
76995.1)8.1('
)8.1(8.1
)('
)(
2
223
f
f
xf
xfxx
We’re already quite close...We’re already quite close...
with much less work than in the bisection method! One more time:
And according to Maple, the root is
fsolve(f(x)=0);
So with not much work we have the answer to six significant figures!
1.769292354
769292663.1)76005.1('
)76995.1(76995.1
)('
)(
3
334
f
f
xf
xfxx
Your turn…Your turn…
Try Newton's method out on the equation
First make a reasonable guess, then iterate. Report your answer when you get two successive iterations to agree to five decimal places.
0325 xx