Second semester summery

This semester is ending soon, and I feel like it went extemely fast.  We learned a lot of things this semester.  We learned probabilities, systems of equations, cramer's rule ....... Honestly, we did not learn a lot of new knowleges, instead we learned a lot of new ways of solving the same type of problems.  Those trick made math easier and more fun. After this semester, I become more comfident about trig problems and probabilities.  I was always bad at probabilities, and I used to never get them right.  But Miss V did a good job explaning and I start to understand it.  I also become better at sequences and series. I didnt perceive well when we first learned it, but when we did the review presentations, I finally understood.  It was a really good idea for us to do presentations as a class to review instead of Miss V go over everything.   This way, we could get more practoce and go over more topics in one day.  If we have questions wecould go ask the person that taught the topic instead of 25 people clumping at Miss V 's desk for answers.  I really like the idea.   This year has been a great year and I hope to finish strong and get a good grade in this class.   The blog posting was intended to help us review weekly, but this semester I could'nt make sure to post on time and that is one thing I regret this year.  

Repeating decimals.

This section, we learned how to write a repeating decimal into a fraction. When we look at a repeating decimal, we can tell that it is the sum of many fractions. For example, 0.4444.....could be written as 0.4+0.04+0.004........  to write the whole thing into a fraction, we have to find the rate.  The rate could be found by diving any consective terms.  In this case is 1/10.  The formula for thia is S= a/1-r.   a represents the first term.   the sum that we get would be this decimal as fraction.  One important thing to remember is that, when r is greater than 1.  The sum do not exist. 

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Tower of hanoi mathmatical induction.

The tower of hanoi is a game/ project we did in class.  in the app on our ipads, we have to use the blocks to move to the third stand in the original order.  After we tried, the app would tell us the actual step we need to get there. Our team tried the game for several times but always takes more steps to complete it. At the end of this game we find an equation show the pattern of this game and can we complete the game perfectly. The equation we prove out is Tn=2^n -1. 
the other lesson was about mathmatical induction.  This looks really complecated at first, but it is actually not when you get it.  First, we need to prove true for n=1, we compare the left and right hand side.  After that, we assume true for n=k, and we need to show true for k+1.  In this step, we replace all the k with k+1. And we simplify.   we rplace the k term with the step when we assume k=n.  And we simplify the problem and we would get same things on both sides and that is thw ultimate goal. 

Trig review week

This week we reviewed trigonometry.  We went over all the trig identities. After reviewing, I became more fimiliar with the identities.  I memorized the most impirtant ones and it helped me with many problems.  For example, the proving type of questions.  Those were exremely hard for me when we first learn it because I never remembered the identities, it was hard for me to recall any when doing the problem so it creates more difficulties. The most troubling part of trig is the double angles for me.  They are still confusing and I hope I can figure them out soon.   the unit circle is very handy when doing trig problems as well, and I am trying to memorize it. 

Parametric equations

Parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter. When given a problem, there are x and y variables and also a parameter t.  In order to eliminate the parameter, we have to use elimination or substitution the do so.  If the equations involves trig, we would have to use the trig identities to help eliminating the parameter.   If we want to graph a parametric equation, we first draw a number box of x, y, and t.  The problems are given a limit of the parameter t, we pick numbers between the limit and we get the values of x and y from equations given.  Last, we plot the points. We have to remember to mark the starting point and the direction of the graph.  And we connect the points. 

Partial fraction

Partial fraction is the process of separating a term into its original form. A+b+c.    when looking at a problem, the first thing to do is to factor the denomenator.   After determining whether the factors are linear or nonlinear. If the term is linear, we put a variable A on top of a factor.  If it is not, we put ax+b.  After this, we multiply the greatest common denomenator to the whole thing.  And we factor out the x s and we set systems of equations to solve for A,B,C.  Finally, we write the answer out.    If the power of denomonator is less than the numerator, we have to do long division first.

Sequence and series.

A sequence is consist with elements and a list of numbers.   a series is a list of numbers adding together.  There are wo types of sequence and series , arithmetic and geometric. Arithmatic means that the elements are related by addition. Geometeic means that the slements are related by multiplication. Arithmetic formula is An= A1+(n-1)d.  A1represents the first term of the series. And d is the difference between two terms. N means the nth term.   Geometric formula is An=A1*r^(n-1)   A1is the first term and r is the rate of two terms.  We can get r by dividing any two terms next to each other. We can write eries  into summation formation.    

Graphing systems of inequalities

Systems of inequalities are similar to the system of regular equations.  It is composed of few equations and it has answers that satisfy the equations.  We first graph the indevidule graphs in the system.  secondly, we need to determine which side of the graph satisfy the inequality.  We can determine this by testing a point on the graph.  The final answer will be the area on the graph which all graphs overlap. 

Cramer's rule

Cramer's rule helps to make solving systems if equations easier.  Because the variables are eliminated and we can just deal with numbers.  First, we put coeficients in matrix form and we take the determinate of that.  And we find Dx or Dy by placing the answer colume to the variable. We take determinate of DY and Dx.  Thus, x=Dx/D and same for other variables.

Systems of equations

When we have a systems of equation, we can solve it algebraically and also solve with geometry we can graph or use substitution/elimination.  When we have exactly one answer, it means the system is consistent and independent. That means the lines intersect at exactly one point. When we have a infinete number of solutions, it means he lines overlap each other so every point on the lines is a solution. That is called consistent nod dependent.  When we have no solution, which is when lines are parallel toeach other. There is no intersection, so there is no solution.  Then the system is inconsistent.   

My polar equation pic

Polar coordinates.

This week we learned about pole cordintes.  Those are points on a circular graph.  It contains a r and a angle value. When graphing, if the angle is positive, we look counterclockwise, and if the angle is negative, we count clockwise. We could switch polar and rectangular coordinates back and force. From polar to rectangular, we use x=rCosø and y=rsinø to get x and y value 

Parabola

This week we talked about prabola. Prabola is a u shape graph either going up and down or left or right. Parabolas have derixtrice.  It also has a focus.  The directrix for vertical graphs is y=k-c and for horizontal graph is x=h-c.  The prabola is symmetric, and there is a axis of symmetry.  The equation for parabola is (x-h)^2=4c(y-k).  If the graph is side to side, the equation's x and y switches.  The focus is (h, k+c) and vertices is (h,k).   

rotating conic

When we graph a conic graph that is tilted, we could try to rewrite it back into the normal equation.  The equations have angles, to rewrite this equations, we first need to find the angle.  Cot2x=A-C/B.    We get the A,B,C values from the equation given.  Ax^2+Bxy+Cy^2+Dx+ey+F=0. He second step is to replace x and y with x'cosø-y'sinø and x!sinø+y'cosø.  The last step is to plug in the x and y into the equation. After plugging in, we just do algebra and simplify the new equation.   And now we have a new equation for the rotated conic graph.  

2nd semester goals

Last semester, I think I did good on the homework part. I was able to submit my works on time and made sure I try for all the questions.  I also kept my notes well and I think it really helped me when I needed to review for tests.  Things that I didn't do so well was i didn't review well before the tests, especially the earlier tests. The informations were too much for me to remember without looking back at the notes for several times.  But when the semester was near ending, I realized I need to review more, and I felt note confident on the tests than before.  For this new semester, I want myself to be more focused on study and make sure to arrange times for reviewing and finish my works on time.  

During my break, I went to Disneyland on Christmas Day, it was fun. There were a lot of people.