Tower of Hanoi-Mathematical Induction

The Tower of Hanoi (also called the Tower of Brahma or Lucas' Tower,[1] and sometimes pluralized) is a mathematical game or puzzle. It consists of three rods, and a number of disks of different sizes which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The objective of the puzzle is to move the entire stack to another rod, obeying the following simple rules: Only one disk can be moved at a time. Each move consists of taking the upper disk from one of the stacks and placing it on top of another stack i.e. a disk can only be moved if it is the uppermost disk on a stack. No disk may be placed on top of a smaller disk. With three disks, the puzzle can be solved in seven moves. The minimum number of moves required to solve a Tower of Hanoi puzzle is 2n - 1, where n is the number of disks.

But you cannot place a larger disk onto a smaller disk. The equation for the tower of Hanoi is 2^n -1. The equation 2^n-1 can be used to solve with any number of discs. Given 12 discs, this would require 4,095 moves.

As disc continue to stack, more moves are required to move them from pole A to C. I also learned that after finding the recursion formula, I was able to plug it into the mathematical induction to make both sides equal.

2nd Semester Summary

It's been really confusing this semester, but it was so much easier than 1st semester. I feel really sad that now I will no longer be with Ms. V. And now I might have to move up to Calc AB and be with Pelletier. I am so scared because my grades were not that grades this year and so that might affect next year, especially knowing how tough Ms. Pelletier might be. D:

Trig Review

In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a2 + b2 = c2" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use.

Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine. sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) 1 + cot2(t) = csc2(t) The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1. sin(–t) = –sin(t) cos(–t) = cos(t) tan(–t) = –tan(t) Notice in particular that sine and tangent are odd functions, while cosine is an even function.

Repeating Decimals

A repeating or recurring decimal is a way of representing rational numbers in base 10 arithmetic. The decimal representation of a number is said to be repeating if it becomes periodic (repeating its values at regular intervals) and the infinitely-repeated portion is not zero. For example, the decimal representation of ⅓ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333…. A more complicated example is 3227 / 555 , whose decimal becomes periodic after the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144…. At present, there is no single universally accepted notation or phrasing for repeating decimals.

Any number that cannot be expressed as a ratio of two integers is said to be irrational. Their decimal representation neither terminates nor infinitely repeats but extends forever without regular repetition. Examples of such irrational numbers are the square root of 2 and pi.

Parametric Equations

In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. are parametric equations for the unit circle, where t is the parameter. 

Together, these equations are called a parametric representation of the curve. A common example occurs in kinematics, where the trajectory of a point is usually represented by a parametric equation with time as the parameter. The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.)

Partial Fraction Decomposition 

Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions. To decompose a fraction, you first factor the denominator. Let's work backwards from the example above. The denominator is x2 + x, which factors as x(x + 1). Then you write the fractions with one of the factors for each of the denominators. Of course, you don't know what the numerators are yet, so you assign variables (usually capital letters) for these unknown values: A/x + B/(x + 1) Then you set this sum equal to the simplified result: (3x + 2)/[x(x + 1)] = A/x + B/(x + 1) Multiply through by the common denominator of x(x + 1) gets rid of all of the denominators: multiply each term by [x(x + 1)] / 1 3x + 2 = A(x + 1) + B(x)

Sequence and Series

A "sequence" (or "progression", in British English) is an ordered list of numbers; the numbers in this ordered list are called "elements" or "terms". A "series" is the value you get when you add up all the terms of a sequence; this value is called the "sum". For instance, "1, 2, 3, 4" is a sequence, with terms "1", "2", "3", and "4"; the corresponding series is the sum "1 + 2 + 3 + 4", and the value of the series is 10. A sequence may be named or referred to as "A" or "An". The terms of a sequence are usually named something like "ai" or "an", with the subscripted letter "i" or "n" being the "index" or counter. So the second term of a sequnce might be named "a2" (pronounced "ay-sub-two"), and "a12" would designate the twelfth term. Note: Sometimes sequences start with an index of n = 0, so the first term is actually a0. Then the second term would be a1. The first listed term in such a case would be called the "zero-eth" term. This method of numbering the terms is used, for example, in Javascript arrays. Don't assume that every sequence and series will start with an index of n = 1.

Graphing Systems of Inequalities

Once you've learned how to graph linear inequalities, you can move on to solving systems of linear inequalities. A "system" of linear inequalities is a set of linear inequalities that you deal with all at once. Usually you start off with two or three linear inequalities. The technique for solving these systems is fairly simple. Here's an example. Solve the following system: 2x – 3y < 12 x + 5y < 20 x > 0 Just as with solving single linear inequalities, it is usually best to solve as many of the inequalities as possible for "y" on one side. Solving the first two inequalities, I get the rearranged system: y > ( 2/3 )x – 4 y < ( – 1/5 )x + 4 x > 0

"Solving" systems of linear inequalities means "graphing each individual inequality, and then finding the overlaps of the various solutions". So I graph each inequality, and then find the overlapping portions of the solution regions.

Cramer's Rule

Given a system of linear equations, Cramer's Rule is a handy way to solve for just one of the variables without having to solve the whole system of equations. They don't usually teach Cramer's Rule this way, but this is supposed to be the point of the Rule: instead of solving the entire system of equations, you can use Cramer's to solve for just one single variable. 

Let's use the following system of equations: 

2x + y + z = 3 

x – y – z = 0 

x + 2y + z = 0

We have the left-hand side of the system with the variables (the "coefficient matrix") and the right-hand side with the answer values. Let D be the determinant of the coefficient matrix of the above system, and let Dx be the determinant formed by replacing the x-column values with the answer-column values:

Systems of Equations 

A system of equations is a set or collection of equations that you deal with all together at once. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables. Think back to linear equations. For instance, consider the linear equation y = 3x – 5. A "solution" to this equation was any x, y-point that "worked" in the equation. 

So (2, 1) was a solution because, plugging in 2 for x: 

3x – 5 = 3(2) – 5 = 6 – 5 = 1 = y 

On the other hand, (1, 2) was not a solution, because, plugging in 1 for x: 

 3x – 5 = 3(1) – 5 = 3 – 5 = –2

Since the two equations above are in a system, we deal with them together at the same time. In particular, we can graph them together on the same axis system.

A solution for a single equation is any point that lies on the line for that equation. A solution for a system of equations is any point that lies on each line in the system. For example, the red point at right is not a solution to the system, because it is not on either line.

Graphing Polar Equations

The most basic method of graphing polar equations is by plotting points and doing a quick sketch. Graphing polar equations is a skill that requires the ability to plot points and sometimes recognize a special case of polar curves, such as cardioids, and roses and conic sections. However, we need to understand the polar coordinate system and how to plot points for graphing polar equations. The graph of a polar equation r = f(θ), or more generally F(r, θ) = 0, consists of all points P that have at least one polar representation (r, θ) whose coordinates satisfy the equation. EXAMPLE: Sketch the polar curve θ = 1. Solution: This curve consists of all points (r, θ) such that the polar angle θ is 1 radian. It is the straight line that passes through O and makes an angle of 1 radian with the polar axis. Notice that the points (r, 1) on the line with r > 0 are in the first quadrant, whereas those with r < 0 are in the third quadrant.
And these are the polar equations that I had created.

Polar Coordinates

The polar coordinates r (the radial coordinate) and theta (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by:

 x = rcostheta (1) y = rsintheta

where r is the radial distance from the origin, and theta is the counterclockwise angle from the x-axis. In terms of √(x^2+y^2) (3) theta = tan^(-1)(y/x). (4) (Here, tan^(-1)(y/x) should be interpreted as the two-argument inverse tangent which takes the signs of x and y into account to determine in which quadrant theta lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, (r,theta+2npi) will be precisely the same polar point as (r,theta) for any integer n. What's more, one often allows negative values of r under the assumption that (-r,theta) is plotted identically to (r,theta+/-pi).

We start with a cartesian coordinate system. We choose O as pole and the x-axis as polar-axis. The cartesian coordinates of a point P are (x,y). Choose the u-axis such that r > 0. then P = r U => P2 = r2 U2 => x2 + y2 = r2 r = sqrt(x2 + y2) Now, choose a t-value such that x = r.cos t and y = r.sin t

Conic Section

Conic sections are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.)

If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles.

There are some basic terms that you should know for this topic: 

Center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola. 

Vertex: in the case of a parabola, the point (h, k) at the "end" of a parabola; in the case of an ellipse, an end of the major axis; in the case of an hyperbola, the turning point of a branch of an hyperbola. 

Focus: a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or "focus". 

Directrix: a line from which distances are measured in forming a conic. 

Axis: a line perpendicular to the directrix passing through the vertex of a parabola. 

Major axis: a line segment perpendicular to the directrix of an ellipse and passing through the foci; the line segment terminates on the ellipse at either end; also called the "principal axis of symmetry"; the half of the major axis between the center and the vertex is the semi-major axis. 

Minor axis: a line segment perpendicular to and bisecting the major axis of an ellipse; the segment terminates on the ellipse at either end; the half of the minor axis between the center and the ellipse is the semi-minor axis. 

Each conic has a "typical" equation form, sometimes along the lines of the following:

parabola: Ax² + Dx + Ey = 0
circle: x² + y² + Dx + Ey + F = 0
ellipse: Ax² + Cy² + Dx + Ey + F = 0
hyperbola: Ax² – Cy² + Dx + Ey + F = 0

Parabola

To form a parabola, you would start with a line and a point off to one side. The line is called the "directrix"; the point is called the "focus". The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola up the middle) is called the "axis of symmetry". The point on this axis which is exactly midway between the focus and the directrix is the "vertex"; the vertex is the point where the parabola changes direction.

The "vertex" form of a parabola with its vertex at (h, k) is: regular: y = a(x – h)2 + k sideways: x = a(y – k)2 + h
The conics form of the parabola equation (the one you'll find in advanced or older texts) is: regular: 4p(y – k) = (x – h)2 sideways: 4p(x – h) = (y – k)2
The relationship between the "vertex" form of the equation and the "conics" form of the equation is nothing more than a rearrangement: y = a(x – h)2 + k y – k = a(x – h)2 (1/a)(y – k) = (x – h)2 4p(y – k) = (x – h)2

                                                                  1st semester 

3 things done well

I didn't do very well first semester. I did well on memorizing the trigonometric identities. I did well on knowing how to solve majority of the trigonometric identities. But for the majority of math, essentially chapters 2-4, I did not do well on. 

3 Goals to improve on

Whenever I need help, I would to come to Ms. V before a test and especially as soon as possible. I need to study more, if I did not get a chance to ask for help. Also I need to resolve my internet issue quicker. 

Favorite Christmas break story

My dad and his friend kept arguing whether to celebrate and eat on Christmas Eve or on Christmas Day.