9. limits
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1) A limit describes the value a function approaches as the input gets closer and closer to a certain value, even if it never exactly reaches it. 2) Geometrically, as a polygon's sides approach infinity, it approaches the shape of a circle but never perfectly reaches it. 3) Numerically, the sequences 1/n and n/(n+1) approach 0 and 1 respectively as n increases without bound. 4) Graphically, the function 1/x approaches 0 as x moves farther from or closer to 0, and a function's limit exists if the left and right-hand limits are equal.
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This document discusses key concepts for limits in calculus, including that a limit represents the intended height of a function at a certain x value, only exists if the same y value is approached from both sides of the graph, and doesn't exist if different y values are approached from the two sides. It also summarizes L'Hopital's rule for evaluating limits of indeterminate forms, and mentions the limit laws and types of discontinuities. The document was produced by Greg, developed by Dan, and presented by Ben.
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Here is a problem I am given Using Diophantus method, find four s.pdf
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their sum added to the sum of their sides is 73. Why does Diophantus\' method always work?
or alternatively, another problem is:
Using Diophantus\' method, show that 73 can be decomposed into the sum of 2 squares in two
different ways. Show that Diophantus\' method always works.
Here is an example of Diophantus\' method, on which I have questions:
Example:
To find four square numbers such that their sum added to the sum of their sides makes a given
number.
Given number 12.
Now x^2 + x + ¼ = a square.
Therefore the sum of four squares + the sum of their sides + 1 = sum of four other squares = 13,
by hypothesis.
Therefore we have to divide 13 into four squares; then if we subtract ½ from each of their sides,
we shall have the sides of the required squares.
Now 13 = 4 + 9 = (64/25 + 36/25) + (144/25 + 81/25),
and the sides of the required squares are 11/10, 7/10, 19/10, 13/10, and the squares themselves
being 121/100, 49/100, 361/100, 169/100.
Where does the x^2+x+1/4 come from?
What does subtracting 1/2 do, where is this evident in the problem?
How would I or anyone else get from (64/25 + 36/25) + (144/25 + 81/25) to 11/10,7/10,19/10,
and 13/10??? This is my biggest question of them all. I would actually like to understand how to
do this problem, so some explanation and an answer as well would be great so I can check the
work myself, and be sure I\'m correct as well.
Solution
The known personal information of Diophantus is best summed up in the following
quote: “Here you see the tomb containing the remains of Diophantus, it is remarkable: artfully it
tells the measures of his life. The sixth part of his life God granted him for his youth. After a
twelfth more his cheeks were bearded. After an additional seventh he kindled the light of
marriage, and in the ?fth year he accepted a son. Elas, a dear but unfortunate child, half of his
father he was and this was also the span a cruel fate granted it, and he consoled his grief in the
remaining four years of his life. By this device of numbers, tell us the extent of his life.” With
our modern notation and use of algebra, it is rather easy to see that if Diophantus lived for x
years, then the problem is equivalent x to solving the equation x + 12 + x + 5 + x + 4 = x, which
has the solution x = 84. 6 7 2 It is hard to pinpoint when exactly Diophantus lived. The Arabian
historian Ab¯’lfaraj mentions u in his History of the Dynasties that Diophantus lived during the
time of Emperor Julian (361 - 363 A.D.), whereas Rafael Bombelli says authoritatively in his
book, Algebra, that Diophantus lived during the reign of Antoninus Pius (138 - 161 A.D.). There
is little or no con?rmation of either of these dates. However, in his On Polygonal Numbers,
Diophantus de?nes a polygonal number by quoting Hypsicles. Hypsicles was the writer of the
supplement to Book XIII of Euclid’s Elements, and so Diophantus must have written aft.
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3) Using the sandwich theorem to evaluate a limit if the function is between two other functions with known limits
It then provides an example using the sandwich theorem to show that limx→0 sin(x)/x = 1 by considering the areas of triangles on a unit circle as x approaches 0.
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Here is a problem I am given: Using Diophantus\' method, find four square numbers such that
their sum added to the sum of their sides is 73. Why does Diophantus\' method always work?
or alternatively, another problem is:
Using Diophantus\' method, show that 73 can be decomposed into the sum of 2 squares in two
different ways. Show that Diophantus\' method always works.
Here is an example of Diophantus\' method, on which I have questions:
Example:
To find four square numbers such that their sum added to the sum of their sides makes a given
number.
Given number 12.
Now x^2 + x + ¼ = a square.
Therefore the sum of four squares + the sum of their sides + 1 = sum of four other squares = 13,
by hypothesis.
Therefore we have to divide 13 into four squares; then if we subtract ½ from each of their sides,
we shall have the sides of the required squares.
Now 13 = 4 + 9 = (64/25 + 36/25) + (144/25 + 81/25),
and the sides of the required squares are 11/10, 7/10, 19/10, 13/10, and the squares themselves
being 121/100, 49/100, 361/100, 169/100.
Where does the x^2+x+1/4 come from?
What does subtracting 1/2 do, where is this evident in the problem?
How would I or anyone else get from (64/25 + 36/25) + (144/25 + 81/25) to 11/10,7/10,19/10,
and 13/10??? This is my biggest question of them all. I would actually like to understand how to
do this problem, so some explanation and an answer as well would be great so I can check the
work myself, and be sure I\'m correct as well.
Solution
The known personal information of Diophantus is best summed up in the following
quote: “Here you see the tomb containing the remains of Diophantus, it is remarkable: artfully it
tells the measures of his life. The sixth part of his life God granted him for his youth. After a
twelfth more his cheeks were bearded. After an additional seventh he kindled the light of
marriage, and in the ?fth year he accepted a son. Elas, a dear but unfortunate child, half of his
father he was and this was also the span a cruel fate granted it, and he consoled his grief in the
remaining four years of his life. By this device of numbers, tell us the extent of his life.” With
our modern notation and use of algebra, it is rather easy to see that if Diophantus lived for x
years, then the problem is equivalent x to solving the equation x + 12 + x + 5 + x + 4 = x, which
has the solution x = 84. 6 7 2 It is hard to pinpoint when exactly Diophantus lived. The Arabian
historian Ab¯’lfaraj mentions u in his History of the Dynasties that Diophantus lived during the
time of Emperor Julian (361 - 363 A.D.), whereas Rafael Bombelli says authoritatively in his
book, Algebra, that Diophantus lived during the reign of Antoninus Pius (138 - 161 A.D.). There
is little or no con?rmation of either of these dates. However, in his On Polygonal Numbers,
Diophantus de?nes a polygonal number by quoting Hypsicles. Hypsicles was the writer of the
supplement to Book XIII of Euclid’s Elements, and so Diophantus must have written aft.
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1. Defining what a limit is graphically as the value a function approaches as the input gets closer to a number without reaching it.
2. Explaining how to find limits algebraically by direct substitution or simplifying expressions.
3. Distinguishing between one-sided and two-sided limits.
4. Identifying infinite limits and discontinuities that can occur with limits involving vertical asymptotes, holes, gaps or jumps in a function's graph.
5. Defining continuity as a function being continuous at a point if it is hole-free, gap-free, and
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6. leidenfrost and galaxy formation
1) The document discusses the Leidenfrost effect and its role in galaxy formation and the debate around antimatter in the universe.
2) It describes criticisms by Gary Steigman of the idea that antimatter exists on a galactic scale, as collisions between matter and antimatter galaxies would produce far more gamma rays than have been observed.
3) The key question is how quickly Leidenfrost layers can form during collisions of matter and antimatter regions, as faster formation would produce fewer gamma rays and less annihilation before the regions bounce apart.
3. plasma galaxies
Plasma currents can organize spiral galaxies without black holes. Computer models simulating interacting electric currents in plasma have reproduced details of spiral galaxies, whereas gravity-only models require hypothetical invisible matter. Laboratory plasma experiments and advanced simulations show electric forces efficiently shape spiral structure through current filaments pinching matter into rotating stars and galaxies, as seen in snapshots of computer simulations reproducing galactic evolution through electric currents.
2. zpe plasma model
The document summarizes a model of the structure of space at very small scales. It proposes that space consists of tiny positively and negatively charged Planck particle pairs that were formed as space expanded. As the particles moved and interacted due to turbulence from the expansion, they generated electric and magnetic fields. The energy from the expansion was ultimately converted into electromagnetic waves as the particles recombined and annihilated each other once expansion ceased and turbulence died down. These electromagnetic waves are considered an intrinsic part of the fabric of space according to the model.
1. introduction
1) The plasma universe model views electro-magnetism as an important force shaping the universe alongside gravity. It sees galaxies and larger structures as forming through plasma filamentation processes rather than solely through gravitational collapse.
2) It proposes that the cosmic microwave background radiation is caused by synchrotron radiation from electrons spiraling around magnetic field lines rather than being leftover radiation from the Big Bang.
3) Stars form within galaxies in this model through the same plasma filamentation and condensation processes as the larger structures, and the model can potentially explain the observed density profiles and expansion of the visible universe through a process involving the separation and explosive interactions of primordial matter and antimatter regions.
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- 1. LIMITS What is a Limit? This lesson explains the concept of a limit (in Calculus) from various points of view. A GEOMETRIC EXAMPLE: Let's look at a polygon inscribed in a circle... If we increase the number of sidesof the polygon, what can you say about the polygon with respect to the circle? As the number of sides of the polygon increase, the polygon is getting closer and closer to becoming the circle! If we refer to the polygon as an n-gon, where n is the number of sides, we can make some equivalent mathematical statements. (Each statement will get a bit more technical.) • As n gets larger, the n-gon gets closer to being the circle. • As n approaches infinity, the n-gon approaches the circle. • The limit of the n-gon, as n goes to infinity, is the circle!
- 2. The n-gon never really gets to be the circle, but it will get darn close! So close, in fact, that, for all practical purposes, it may as well be the circle. That's what limits are all about! Archimedes used this idea (WAY before Calculus was even invented) to find the area of a circle before they had a value for PI! (They knew PI was the circumference divided by the diameter... But, hey, they didn't have calculators back then.) SOME NUMERICAL EXAMPLES: EXAMPLE 1: Let's look at the sequence whose nth term is given by n/(n+1). Recall, that we let n=1 to get the first term of the sequence, we let n=2 to get the second term of the sequence and so on. What will this sequence look like? 1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,... 99999/100000,... What's happening to the terms of this sequence? Can you think of a number that these terms are getting closer and closer to? Yep! The terms are getting closer to 1! But, will they ever get to 1? Nope! So, we can say that these terms are approaching 1. Sounds like a limit! The limit is 1. As n gets bigger and bigger, n/(n+1) gets closer and closer to 1... EXAMPLE 2: Now, let's look at the sequence whose nth term is given by 1/n. What will this sequence look like? 1/1, 1/2, 1/3, 1/4, 1/5,... 1/10,... 1/1000,... 1/1000000000,...
- 3. As n gets bigger, what are these terms approaching? That's right! They are approaching 0. How can we write this in Calculus language? What if we stick an x in for the n? Maybe it will look familiar... Do you remember what the graph of f(x)=1/x looks like? Keep reading to see our second example shown in graphical terms! SOME GRAPHICAL EXAMPLES: On the previous page, we saw what happened to the sequence whose nth term is given by 1/n as n approaches infinity... The terms 1/n approached 0. Now, let's look at the graph of f(x)=1/x and see what happens! The x-axis is a horizontal asymptote... Let's look at the blue arrow first. As x gets really, really big, the graph gets closer and closer to the x-axis which has a height of 0. So, as x approaches infinity, f(x) is approaching 0. This is called a limit at infinity. Now let's look at the green arrow... What is happening to the graph as x gets really, really small? Yep, the graph is again getting closer and closer to the x-axis (which is 0.) It's just coming in from below this time.
- 4. But what happens as x approaches 0? Since different things happen, we need to look at two separate cases: what happens as x approaches 0 from the left and at what happens as x approaches 0 from the right: and Since the limit from the left does not equal the limit from the right... Let's look at a more complicated example... Given this graph of f(x)...
- 5. First of all, let's look at what's happening around the dashed blue line. Recall that this is called a vertical asymptote. So... Another way to think about the limit is the find the height of the graph at (or really close to) the given x. Think about a little mountain climbing ant (call him Pierre) who is crawling on the graph. What is Pierre's altitude when he's climbing towards an x? That's the limit! Let's try some more... Let's look at what's happening at x = -7... The limit from the right is the same as the limit from the left... But there's a hole at x = -7!
- 6. That's ok! We don't care what happens right at the point, just in the neighborhood around that point. So... Can you find the limit of f(x) as x approaches -3? That's right! How about the limit of f(x) as x approaches 0? Right again! Let's look at one more type of limit. To do this we'll show you the screen of a TI-92 graphing calculator! This is the graph of (If you have a TI-92, the viewing window here is -1.5, 2.1, 1, -3.3, 4.7, 1, 2.) It sure wiggles around a lot! But, we see that