AUGUST 2

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AUGUST 2
Limits, sequences, and improper integrals
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MATH 104Calculus I
Review of previous material. methods and
applications of integration, differential
equations ..
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Geometry of Differential Equations
A differential equation of the form
gives geometric information about the graph of
y(x). It tells us
If the graph of y(x) goes through the point
(x,y), then the slope of the graph at that point
is equal to f(x,y).
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An example
We can draw a picture of this as follows. For the
differential equation
, we have
If the graph goes through (2,3), the slope must
be 1 there. If the graph goes through (0,0), the
slope must be 0 there. If the graph goes through
(-1,-2), the slope must be -1 there.
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Put it on a graph...
The slope of the arrow at any point is equal to y
- x at that point. This kind of picture is called
a "direction field" for the differential equation
dy/dx y - x . We can use this to solve the
differential equation geometrically and recover
the graph of the function.
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The idea is to start
The idea is to start somewhere on the direction
field and simply follow the arrows
This graphical technique is useful for getting
qualitative information about solutions of
differential equations, especially when they
cannot be integrated.
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Here are a couple for you to try...
y ' 2 ( y - y2)
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y ' 3 x sin(2y)
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Numerical methods
Another way to gain insight into solutions of
differential equations is to use numerical
methods for their solution. The simplest
numerical method is called Euler's method.
Euler's method is easy to understand if you
relate it to two things you already know 1. The
left endpoint (rectangle) method for estimating
integrals, and 2. The fundamental theorem of
calculus. Or, you can think of Euler's method
in terms of differentials
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Eulers method
You can algebraically manipulate most first-order
equations, until they are in the form y'(x)
f(x,y) Euler's method then combines the
differential formula with the differential
equation In Euler's method, we simply ignore
the small errors and repeatedly use the resulting
equation with a small value of to
construct a table of values for y(x) (that can
then be graphed, for instance).
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An example...
y ' y - x, y(0) 2 (this is the example we
graphed before). We'll use Dx 0.1 The choice
of Dx is usually dictated by the problem or the
situation. The smaller Dx is the more accurate
the approximated solution will be, but of course
you need to do more work to cover an interval of
a given length. For the first step, we can use
that x0 and y2, therefore y ' 2. Euler's
method then tells us that y(x Dx)
y(x) f(x,y) Dx
y(0.1) 1 (2 - 0) 0.1 1.2
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Continue...
For the second step, we have x 0.1, y 2.2,
therefore y' 2.1. Euler's method then gives
y(0.2) 2.2 2.1(0.1) 2.41 We continue in
this manner and fill in the following table
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Maple...
Maple tells us that the exact solution of the
equation y ' y - x that has y(0) 2 is
y(x) x 1 ex and so we have y(1)
4.718281828. So the Euler method result is
pretty close (within 10). We could do better by
decreasing Dx, but of course then we'd need more
steps to reach x 1. You'll get to try a couple
of these on this week's homework.
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LHospital
Another useful technique for computing limits is
L'Hospital's rule Basic version
If
, then
provided the latter exists. This also applies if
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Fancy LHospital
You can use Basic LHospital for 4, 6, 8 and
10 of the top ten limit list. But for limits like
9, you need...
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Top ten famous limits
1.
2.
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3. (A) If 0 lt x lt 1 then
(B) If x gt 1, then
4. and
5. and
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6-10
6. For any value of n, and for any
positive value of n,
7.
does not exist!
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8.
9.
10. If f is differentiable at a, then
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Here are three more
A challenge
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How about this one?
A. B. 0 C. - D. 1
E. F. G. H.
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Last one (for now)...
A. 0 B. 1/2 C. D. 3
E. F. undefined G. H.
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Improper integrals
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.
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Examples of the first kind are
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Examples of the second kind
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.
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Same method
No matter which kind of improper integral (or
combination of improper integrals) we are faced
with, the method of dealing with them is the
same
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Calculate the limit!
What is the value of this limit (and hence, of
the improper integral )?
A. 0 B. 1 C. D. E.
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Another improper integral
A. 0 B. C.
D. E.
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Area between the x-axis and the graph
The integral you just worked represents all of
the area between the x-axis and the graph of
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The other type...
For improper integrals of the other type, we make
the same kind of limit definition
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Another example
What is the value of this limit, in other
words, what is
A. 0 B. 1 C. 2 D. E.
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A 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
so this improper integral diverges.
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One for you
A. Converge B. Diverge
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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).
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A puzzling example...
Consider the surface obtained by rotating the
graph of y 1/x for x gt 1 around the x-axis
Lets calculate the volume contained inside the
surface
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What about the surface area?
This is equal to...
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".
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Sequences
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,
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Convergent and Divergent
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
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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.
n
2
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Practice
The sequence
A. Converges to 0 B. Converges to 1 C.
Converges to n D. Converges to ln 2 E. Diverges
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Another...
The sequence
A. Converges to 0 B. Converges to 1 C.
Converges to -1 D. Converges to ln 2 E. Diverges
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A powerful existence theorem
It 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 lta lta lt .... 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
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Consider the sequence...
etc.
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 xgt2, 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.
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QUESTION
What is the limit?
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Newtons Method
A 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 y0, we solve for x in
the equation f(a) f '(a)(x-a) 0 and get x a
- f(a)/f '(a).
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Lets 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
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Were 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
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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.
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Fractals
Fractals are geometric figures constructed as a
limit of a sequence of geometric figures. Koch
Snowflake Sierpinski Gasket Newton's method
fractals
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Good night!
See you on Wednesdy Check the web! Turn in
homework! Have a great week!