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CALCULUS II EXAM III NOTES AND LINKS |
| If it is not already on your hard drive, you will need to download
the free DPGraph Viewer to view some of
the pictures linked to on this page. |
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QuickTime free download. |
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9.1 You need to be able to determine
whether or not a sequence converges and if it does converge be able to find its
limit. You need to be able to recognize some common sequence patterns and
given the first few terms in the sequence be able to write an expression for the
nth term of the sequence that at least works for the given first few terms.
| To the right is a plot of the sequence an = (8n3
- 4n2)/(n3 + 20n + 10) as n goes from 1 to 50. What
would the corresponding infinite sequence converge to? |
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9.2 You need to distinguish between the
term "sequence" and "series". You
need to understand and be able to use summation notation. You need to
understand the terminology and definition of convergent and divergent series
given on page 606. You need to be able to recognize a telescoping series,
determine whether or not it converges, and compute what it converges to if it
does converge. You need to be able to recognize a geometric series and be
able to sum (determine what it converges to) a convergent infinite geometric
series. You should also be able to sum any finite geometric series.
You need to know and be able to apply theorems 9.8 and 9.9. Be able to
prove that the sphereflake , P615:108, has an infinite surface area. Here
are some drill
problems on geometric series, the nth term test, and telescoping series.
You should be able to compute the total distance traveled by an idealized
bouncing ball dropped from a given initial height that always bounces back up to
some fraction (less than one) of the height it just fell from. The first
five people to correctly identify the fraction (ratio r) in this bouncing
ball animation will receive a bonus.
an
= 10(3/4)n |
The picture on the left shows a graph of the
first 51 terms of a geometric sequence and the picture on the right shows
a graph of the sequence of partial sums of the corresponding geometric series.
What does the geometric sequence converge to? What does the infinite geometric
series converge to?
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Sn = 40(1 - (3/4)n) |
9.3 You need to be able to apply the
Integral Test to appropriate infinite series to prove that the series either
converges or diverges. We will also discuss how to extend the Integral
Test in order to put a bound on the error in summing a finite number of terms in
an attempt to approximate the value a convergent infinite series converges
to. Here are some drill
problems dealing with the Integral Test. You need to know what a p-series is, what a harmonic series is, and
how to apply Theorem 9.11.
9.4 You need to be able to apply the Direct
Comparison Test and the Limit Comparison Test to determine whether certain types
of infinite series converge or diverge. Here are some drill
problems on the Limit Comparison Test.
9.5 You need to know what an alternating series
is. You need to be able to apply the Alternating
Series Test to determine convergence of series to which the test applies and to
be able to put a bound on the error in summing the first n terms of a convergent
alternating series by using the Alternating Series Remainder Theorem (P
633). You need to be able to approximate the sum of a convergent
alternating series to a specified level of accuracy by using your calculator or
computer. Here are some drill
problems dealing with alternating series. You need to be able to determine whether a convergent series
converges absolutely or conditionally.
9.6 You need to be able to apply the Ratio
Test and the Root Test to test for absolute convergence. You will not
have to use the Root Test since the Ration Test will handle all the examples we
do in class. Here are some drill
problems on the Ratio Test. Here are more
drill problems focusing on trying to determine which test to use when
testing for convergence.
Here is the HMC
summary of convergence tests for infinite series. Here is a link that
includes some convergence tests and some common
series expansions.
| 9.7 You need to be able to construct Taylor
and Maclaurin polynomials to approximate given functions. You need to be
able to apply Taylor's Theorem (P 654) to put a bound on the error in using a
Taylor Polynomial to approximate the value of a given function for a given value
of the independent variable (usually x). Here is an applet that will graph
Taylor approximations to
the sine function and another one that will graph Taylor
approximations to exp(x). Here is a nice Quicktime
animation that looks at Maclaurin polynomials approximating f(x) =
sin(x) + cos(x). Below are the first, third, fifth,
seventh, ninth, eleventh, and thirteenth Maclaurin polynomials
approximating the sine function (sine function in blue). |
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Here is a short PowerPoint
presentation showing the 1st through the 23rd Maclaurin polynomials
approximating the sine function and a slightly
longer version.
9.8 You need to be able to recognize the
type of infinite series that would be called a power series centered at
c. You need to be able to find the domain of a function defined by a power
series. This means you will need to be able to find the radius of
convergence and interval of convergence of a power series including being able
to check for endpoint convergence. You need to know the conditions under
which you can differentiate and integrate power series and be able to do it.
| 9.9 You will need to be able to use the formula
for summing a convergent infinite geometric series to construct the power series
representation for a variety of functions. You will need to be able to use
"Operations with Power Series", page 671, to construct power series
representations for functions from the power series representations of other
functions. You will need to be able to apply the technique of finding a
power series for a given function by integrating the power series for another
function. To the right is the graph of y = ln(x) in blue and the graph of
the Taylor polynomial P6(x) centered at 1 approximating ln(x)
in red.
P6(x) = (x-1) - (1/2)(x-1)2
+ (1/3)(x-1)3 - (1/4)(x-1)4 + (1/5)(x-1)5
- (1/6)(x-1)6
What would be the interval of convergence of the
corresponding infinite series?
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| 9.10 You need to know the definition of
Taylor and Maclaurin series. You need to know the meaning of Theorem 9.23,
Convergence of Taylor Series. In particular you need to understand Rn(x)
(some call this Rn+1(x)) in Taylor's
Theorem. You need to be familiar with the power series representations
for elementary functions shown on page 682 of your text
and be able to use these results to construct power series representations of
other functions. You need to be able to use power series to approximate
definite integrals that you would otherwise not be able to evaluate using the
Fundamental Theorem of Calculus. At the right is the graph of y = ex
in blue and the Maclaurin polynomial P6(x) approximating ex
in red.
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You need to be able to compare on your graphing
calculator or computer the graphs of a given function, say sin(x), and its nth Taylor or
Maclaurin polynomial, Pn(x), and be able to determine graphically an
interval over which the function values of sin(x) and Pn(x) differ by
less than a prescribed amount, say 0.00001.
| Zeno's Paradox
The Greek philosopher Zeno (495 - 435 B.C.) may
have been a fast runner at one time and perhaps claimed that if he was
given a slight head start no one on earth could catch him. If so
then by the paradox named after him he would still have made the claim
even after he aged and was no longer a fast runner. This is because,
as he said, to catch him one must get to where he was and by then he will
have moved forward. You can never get to where he is because to get
to where he is you must get to where he was first and by the time you get
to where he was he will no longer be there. This implies that to
score a touchdown in American football a wide receiver has merely to catch
the ball behind the secondary. No matter how fast the defensive
backs are they will not be able to catch up to the receiver and the
receiver can simply walk to the end zone, taking care never to pause along
the way. As long as he never stops moving and continues heading
toward the end zone no one can ever reach his position from behind
him. He would have to stop moving away from the quarterback briefly
to catch the football since otherwise the football could never catch up to
him.
To present this paradox mathematically, Zeno
considered the case of Achilles and the tortoise. I will consider an
aged Achilles and the tortoise although the same argument would apply to a
young fleet Achilles and the tortoise. Suppose Achilles runs at a
rate of 1 meter/sec and the tortoise crawls at a rate of 9/10
meter/sec. If the tortoise is given a 1 meter head start and
continues to travel at a rate of 9/10 m/sec then Achilles traveling at 1
m/sec cannot catch up to him. This is because once Achilles has
traveled the 1 meter to get to where the tortoise was when the chase
began, the tortoise will have traveled 9/10 of a meter farther. Once
Achilles travels that 9/10 of a meter the tortoise will have traveled an
additional (9/10)2 meter or 81/100 meter. Once Achilles
travels this 81/100 meter the tortoise will have traveled an additional
(9/10)3 meter or 729/1000 meter, and so on so Achilles will
never catch the tortoise. Achilles will travel a distance of
and in spite of n approaching infinity he will
never catch the tortoise. Can you mathematically demonstrate the
flaw in this argument?
In the animation
of this paradox the tortoise is in blue, Achilles in red, and the time
that has elapsed during the chase is shown in green along the y-axis.
Quicktime animation |
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