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Sequences and Series           Ayush Mathur XI B

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Sequences and Series           Ayush Mathur XI B

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What is a sequence?

Here are a few lists of numbers:

· 21, 16, 11, 6 ...

· 1, 2, 4, 8 ...

Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term.

| 3, | 5, | 7,…
| 1st term | 2nd term | 3rd term

Sequences usually have patterns that allow us to predict what the next term might be.

Series.
Let a1,a2,a₃, an… be a given sequence. Then, the expression

a1 + a2 + a₃ +,…+ an +…

is called the series associated with the given sequence.

Arithmetic progression

A sequence of numbers in which each differs from the preceding one by a constant quantity (e.g. 1, 2, 3, 4, etc.; 9, 7, 5, 3, etc.).

Geometric progression

A progression of numbers with a constant ratio between each number and the one before (e.g. 1, 3, 9, 27, 81).

Given two numbers a and b. We can insert a number A between them so that a,A,b is an Arithmetic Progression. This number A is called the Arithmetic Mean of the two numbers a and b.
A = (a+b)/2
The Geometric Mean of two positive numbers a and b is the number root of ab.

CCL:-  HISTORY
Evidence is found that Babylonians, some 400 years ago, knew of arithmetic and geometric sequences
SCIENCE
On plotting straight line graphs for relations between two or more functions/formulas/units, AP can be found and used to find distinct points on the graph without actually using the X and Y axis denotations.

WL:- https://brilliant.org/wiki/geometric-progressions/#applications

https://www.quora.com/What-are-some-examples-of-Arithmetic-Progression-in-Nature

https://matheducators.stackexchange.com/questions/11106/examples-of-arithmetic-and-geometric-sequences-and-series-in-daily-life

https://www.youtube.com/watch?v=AjAO0myllzw

https://www.youtube.com/watch?v=O7x3LBWn-Ao

https://www.youtube.com/watch?v=SjSHVDfXHQ4  (personal favorite)

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- Jeanne Liz, 11 B
World Link :
https://www.quora.com/What-are-some-real-life-applications-of-set-theory
https://www.quora.com/What-is-the-use-of-sets-in-real-life
https://www.quora.com/What-practical-application-does-set-theory-have-in-the-real-world
https://www.quora.com/How-can-set-theory-be-used-in-real-life-situations
https://prezi.com/5ni7vvlqc2rm/set-theory-in-real-life/
CCL Link :
https://www.quora.com/What-are-some-possible-applications-of-set-theory-to-physics
https://link.springer.com/article/10.1007%2FBF02478351
https://www.slideshare.net/farzumimu/applications-of-set-theory-in-economical-problem (9th slide onwards)
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2664335
http://smallbusiness.chron.com/sets-set-theory-business-operations-36927.html

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What is Conic Section?
A figure formed by the intersection of a plane and a circular cone. A conic section may be a circle, an ellipse, a parabola, or a hyperbola.

What are the uses of conic section in real life?
Here are some real life applications and occurrences of conic sections:

1. Parabolic mirrors are used to converge light beams at the focus of the parabola.

2. solar ovens use parabolic mirrors to converge light beams to use for heating.

3. The parabola is used in the design of car headlights and in spotlights because it aids in concentrating the light beam

4. Hyperbolas are used in a navigation system known as LORAN (long range navigation)

5. Hyperbolic as well as parabolic mirrors and lenses are used in systems of telescopes
etc.

Why are conic sections so important?
Conic sections are very important because they are useful in studying 3D geometry which has wide applications . In electro magnetic field theory it helps us study the nature of the field inside different shapes of conductors. Knowledge on conic sections is required for designing antennas like conical antenna, pyramidal antenna, parabolic reflectors etc . In mechanical engineering it helps in creating engineering drawing which is the basic plan of components designed.

Kindly visit the following link to know more about conic sections:
https://www.khanacademy.org/math/precalculus/conics-precalc#conic
section-intro

Questions:
1.Which type of conic section has an eccentricity of 1?
2.What are the vertex, focus, directrix, and line of symmetry for 8(y – 2) = (x + 1)²?

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Mathematical Reasoning - Mahalakshmi Kannan - 11A

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Mathematical Reasoning - Mahalakshmi Kannan - 11A

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Some of the reasonings are used in physics, chemistry, and computer science also.
*https://youtu.be/SOxeHyXl3vA

*https://schools.aglasem.com/46113

*https://www.askiitians.com/revision-notes-new/maths/mathematical-reasoning/

*https://gupea.ub.gu.se/bitstream/2077/38715/1/gupea_2077_38715_1.pdf

*https://www.teachingchannel.org/videos/real-world-math-examples

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Permutations and Combinations-                                                                  ISHIKA GOEL XI B.

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Permutations and Combinations-                                                                  ISHIKA GOEL XI B.

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WL:-
*https://betterexplained.com/articles/easy-permutations-and-combinations/
*https://revisionmaths.com/advanced-level-maths-revision/statistics/permutations-and-combinations
*https://www.qsleap.com/gmat/resources/crash-course-mathematics-permutations-and-combinations-part-1
*https://www.mathsisfun.com/combinatorics/combinations-permutations.html

CCL:-
*https://www.khanacademy.org/math/precalculus/prob-comb/combinations/e/permutations_and_combinations_2
*http://www.cpalms.org/Public/PreviewResourceLesson/Preview/71610
*http://www.transum.org/Software/SW/Starter_of_the_day/Similar.asp?ID_Topic=8

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The binomial theorem (or binomial expansion) is a result of expanding the powers of binomials or sums of two terms. The coefficients of the terms in the expansion are the binomial coefficients . The theorem and its generalizations can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics.

The binomial theorem generalizes special cases which are common and familiar to students of basic algebra:

(x+y)^1 = x+y

(x+y)^2 = x^2 + 2xy + y^2

(x+y)^3 = x^3 + 3x^2y+3xy^2+y^3

(x+y)^4 = x^4 + 4x^3y + 6x^2y^2+4xy^3+y^4

The binomial theorem also helps explore probability in an organized way:

Example Question: A friend says that she will flip a coin 5 times. Each time the coin comes up heads, she will give you $10, but each time the coin comes up tails, she gives nothing. What is the probability that you will win $30 playing this game?

XXX

Real World Examples of Binomial Theorem

Distribution of IP addresses

National economic prediction
Architecture
Weather forecasting
Ranking of candidates

From <https://www.quora.com/What-are-some-real-world-examples-of-the-use-of-the-binomial-theorem> & <https://brilliant.org/wiki/binomial-theorem-n-choose-k/>

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Binomial theorem | Polynomial and rational functions | Algebra II | Khan Academy

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limits and derivations anjali singh 11-a 

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limits and derivations anjali singh 11-a 

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some links you can refer for better understanding of the concepts:
https://www.youtube.com/watch?v=jRxr30s-QBU

https://www.youtube.com/watch?v=HTd_6eSdljQ

https://www.vedantu.com/revision-notes/cbse-class-11-maths-notes-chapter-13-limits-and-derivatives

https://www.scribd.com/doc/130108602/Class-11-Maths-Important-Questions-Limits-and-Derivatives

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PRINCIPLE OF MATHEMATICAL INDUCTION-PRANAY BARTHWAL - 11 A

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PRINCIPLE OF MATHEMATICAL INDUCTION-PRANAY BARTHWAL - 11 A

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MATHEMATICAL INDUCTION

The  Principle of mathematical induction

THE NATURAL NUMBERS are thecounting numbers:  1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

Suppose we have some statement P,n and we want to demonstrate that P,n is true for all n � �. Even if we can provide proofs for P�0, P�1, ..., P�k, where k is some large number, we have accomplished very little. However, there is a general method, the Principle of Mathematical Induction.

Example Question:-

1. There is a tower of nn disks, stacked in decreasing size on one of 33 pegs.

How many moves are needed to move all the disks onto a different peg?

Real World Example Of Mathematical Induction:-

From:-

<https://www.quora.com/What-is-the-use-of-Mathematical-Induction-in-real-life>

<h ttps://www.quora.com/What-are-the-real-life-examples-of-the-principle-of-mathematical-induction >

<https://youtu.be/ZpPk99D_Vlg>

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Inequalities can be used to model a number of real-life situations. When converting such word problems into inequalities, begin by identifying how the quantities relate to each other, and then pick the inequality symbol that is appropriate for that situation. When solving these problems, remember that the solution will be a range of possibilities—inequalities do not have a single answer, as equations do. Absolute value inequalities can be used to model situations where margin of error is a concern.

https://www.youtube.com/watch?v=UTZq73wdcqc
Susan earns $80 per day and $12/ hour for overtime. If she cannot earn more than $130 in a day, what is the maximum number of hours she can work overtime?

A taxi cab charges a fixed amount of $1.50 in addition to $0.75 per mile. If Jasmine has $20 with her, what is the maximum number of miles she can ride in the cab?

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Principle Of Mathematical Induction - Pranay Barthwal

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Principle Of Mathematical Induction - Pranay Barthwal

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MATHEMATICAL INDUCTION

The principle of mathematical induction

THE NATURAL NUMBERS are the counting numbers:  1, 2, 3, 4, etc. Mathematical induction is a technique for proving a statement -- a theorem, or a formula -- that is asserted about every natural number.

Suppose we have some statement P,n and we want to demonstrate that P,n is true for all n � �. Even if we can provide proofs for P�0, P�1, ..., P�k, where k is some large number, we have accomplished very little. However, there is a general method, the Principle of Mathematical Induction.

Example Question:-

There is a tower of nn disks, stacked in decreasing size on one of 33 pegs.

How many moves are needed to move all the disks onto a different peg?

Real World Example Of Mathematical Induction:-
From:-
<https://www.quora.com/What-is-the-use-of-Mathematical-Induction-in-real-life>

<https://www.quora.com/What-are-the-real-life-examples-of-the-principle-of-mathematical-induction>

<https://youtu.be/ZpPk99D_Vlg>

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limits and derivatives                    Anjali Singh 11-A

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limits and derivatives                    Anjali Singh 11-A

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Limits
In the study of calculus, we are interested in what happens to the value of a function as the independent variable gets very close to a particular value.
derivatives
To understand the topic derivatives let us take an example of throwing a ball straight up in the air.  Because gravity acts on the ball it slows down, then it reverses direction and starts to fall.
Now, let's zoom in on the section of the graph near t=1
Observing the graph, we see that it passes through
slope= 29m/s

The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.

Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve.