sorry this is so late.... two weeks ago in class we learned about probability. We learned about sample spaces and focused on probability using dice and cards.

We started class today by watching a Formula 1 Racing video, which was really cool but which also led to one of the topics we covered in class- velocity. We realized that it at one point in time, an object is not moving (think of a snapshot: in a picture, an object is stationary). The solution to that problem was the term instantaneous velocity.

In addition to this, we learned about finding the slope of the line tangent to a given graph at a certain point, also called the derivative, and finding the formula for the slope of a line tangent to a given graph at any point on the graph.

Limits with piecewise functions, One-sided limits, and More Algebra with Limits

We started class today by going over the homework from the night before (4/30). We then started talking about the limits of piecewise functions and one-sided limits, which is when you determine the limit of something by looking at it from either the positive or negative direction. This is indicated by having a + or - superscript. We then had more examples of algebra with limits.

More Limit Examples

 Today in class we went over more examples of limits. We focused mainly on how to approach the problem if  you get the fraction (0/0) when you first "plug in" the limit number.  If you do get (0/0) then you have to factor in order to cancel out like terms.  The "secret" that we learned today, was that you solve derivatives the same way you solve these limit problems. 

When a limit does not exist; Algebra and Limits; Direct Substitution

Today we learned that 3 reasons that a limit does not exist: the limit does not exist because of oscillating behavior; the limit does not exist because of unbounded behavior; the limit does not exist because of different left/right behavior.  We learned that algebra and limits is "nice" because it makes sense to us.  Direct substitution involves plugging the limit number into the equation.

Evaluating Limits

So today we started limits. We use limits as "mathematical microscopes" to describe y-values near a given x-value. We evaluate the limits of given functions by (for now) using the calculator: Enter the function in Y1 and use the table feature to enter certain x-values. Use the y-values to determine the limit of the function. Remember to use radians when evaluating limits that involve trig. Also, keep in mind that some limits do not exist or may be deemed "indeterminate." See notes for further details.

Binomial Theorem

Today we learned about the binomial theorem. We learned the real formula on how to expand (x+y)^n. We learned the combination function and reviewed Pascal's Triangle, as both can be used to expand (x+y)^n. You can use Pascal's Triangle to find the coefficients of term. 

Proof by Induction

Sum Equations

Today in class we learned equations to find the sums of certain types of sequences.

Geometric Sequences

Monday in class we talked about geometric sequences, where there is a constant ratio between terms. We learned the equation and worked some practice problems, as usual. We also discussed finding the sum of terms in a geometric sequences, which usually has to be done on a calculator.

Arithmetic Sequence

new formula for linear functions, new term Arithmetic Sequence 

d is the constant difference between the terms 

can find the formula from a sequence, can also find a sequence from a formula 

lots of examples 

Sequences, Recursion, Sigma

Today we practiced sequences, recursive notation, and played with summation/sigma. The new things were mainly the latter 2. Recursive notation is a way of writing a sequence in such a way that the method calls itself. Recursive notation requires 2 things: An defined by a change in An-1 and a start point, a value for A1. Sigma notation is used when talking about adding everything up in a sequence. You put the lower limit on the bottom and the upper limit on the top. Next to the sigma, write the function that you will call each time to produce the next number in the sequence and that number will get added to the previous number until the upper liimit has been reached. You can use the calculator if the numbers get messy. 2nd -> Stat -> math for sum and -> ops for seq and then type in the function, variable being used, start value, end value

Sequences

Yesterday's class involved an introduction to sequences - namely, continuing a few sequences and writing formulae for them.

Square Roots of Complex Numbers

In class, we learned how to find the square roots of complex numbers.  We also learned how to take the cube and fourth roots of complex numbers by solving some sample problems.  

Complex Number Arithmetic

Today in class, we learned about complex number arithmetic. We learned a new notation involving cis instead of (cos + i sin). Also, we learned how to multiply and divide complex numbers. Finally, we learned Moivre's Theorem when dealing with powers and complex numbers.

Polar Graphing

Today in class we learned about cardioid and limacon types of polar graphs and we graphed examples on a worksheet. We talked about how petals of a rose curve are on the axis when there is a cosine equation. We also learned that limacon curves have an inner loop, they are dimpled, and are convex.

Polar Graphing

Today in class, we learned how to graph polar equations. We started by plotting points on polar graph paper. We graphed two different examples, a circle and a rose curve. Then, we studied how different parts of the polar equation affect the graph.

Converting Between Polar and Rectangular Equations

In class, we learned how to convert between polar and rectangular equations, mainly using substitution. Then we did some examples on the board just to see the different ways to convert between the two; some more sneaky then others.

Polar Graphing

Today we started class talking about how you need to be good at 30, 60, 90 and 45,45,90 triangles as well as being able to think even when things are not intuitive. (see first page of notes, shortest path is unexpected.)

We learned that a coordinate on a polar graph is labeled (distance, angle). Points on polar graphs have an infinite number of names. We also learned the formulas to convert polar points to rectangles points and vice versa. (this is all on the second page)

Plus we did a few examples. (see 3rd page)

Notes for February 17

We learned the basic parametric equations for circles, ellipses, and hyperbolas. We also did a problem using parametric equations to chart the path of a projectile.

February 16 Class Notes

Today in class, we discussed the different ways to write parabolas (vertex/standard form), learned how to find the focus and directrix, and drew graphs of parabolas.

More hyperbola and ellipse

Today in class we discussed more facets of ellipses and hyperboles. In other words, we learned how to write equations and draw both kinds of figures when given only a certain amount of starting information (ex. verticies and foci, center and foci, asymptotes, etc.).

-Also, we learned that "a" and "b" simply represent distances from the center of the figure.

-Where the asymptotes intersect on a hyperbola is the center of the hyperbola.

This is mostly useful information for our quiz on Thursday, Feburary 12th.

Conic Sections: Hyperbolas

In class today, we reviewed out second conic section: hyperbolas.

Hyperbolas act like the opposite of ellipses. Rather than adding the x and y terms in the standard form of the equation, we subtract them. Rather than subtracting a^2 and b^2 to find the foci, we add them.


We reviewed the "box" method of graphing in which we can draw a rectangle centered at (h,k) with length 2b and width 2a. This can be used to draw the desired asymptotes/vertices in the graph.

Pay close attention to the point-slope method to find the asymptotes and the way we decide what direction a hyperbola goes.


Finally, we reviewed how to change a messed-up equation in the form:

Ax^2+Bx+Cy^2+Dy+E=0

into standard form at the end of class(where A or C are negative).





Whenever you want to graph a hyperbola: label vertices, foci, asymptotes(show box work)

Monday Feb 9

We covered ellipses in class today. I am not going to post any notes today - instead I will just direct your attention to p640-641 in your textbook.

We will cover hyperbolas tomorrow and then review both on Wednesday. The quiz on Thursday will be over only those two shapes, as we will get to parabolas next week.

Class on Feb 5

There was a quiz today covering vectors, vector operations and vector applications.

There are no new notes today and the homework is a worksheet. The worksheet introduces vectors of the form <>.

notes Feb 4

Here are my notes for today. They were a little messier than usual, so I upped the scan quality just a bit.

The main concept is the dot product. The other big idea is the formula for finding the angle between two vectors.

There is a quiz tomorrow.

We went over three example problems in class today. Each is an application of vectors and could be considered "physics problems" as much as "pre-cal problems."

Note you can leave answers in component form unless specifically asked for the magnitude and/or direction of the vector.

notes for Feb 2

The only new concept I introduced in class today was the idea of the unit vector.

I did one physics-type example problem with velocity. The vector idea was that the velocity was given as magnitude and direction, and you need to break it down into horizontal and vertical components. Recall too that we have the old "projectile motion" model equation from our unit on quadratics.

We will go over several other physics-type problems in class tomorrow, including navigation and tension problems

first!

So welcome to your precal blog.

To begin with, this will be a place where we will keep track of what happened in class each day. One student will be responsible for taking good notes and then posting them here along with a recap of the days activities and themes. As there are 43 of you, you will only need to be the "note-taker-of-the-day" at most twice.

This will not be a graded activity, but it is expected that you will contribute when called upon. As this is new for me, please understand that format, style, and expectations may change as we go.