May 5 - Shortcuts to the Derivative

If the main concept yesterday (Tuesday) was the definition of the derivative, today's main idea was how to best calculate a derivative. It is important that you always keep in mind that the derivative is defined to be the limit of the slope between points, but we will rarely (never) use the limit definition to find the derivative of a function.

For now it's all about the Power Rule. Examples are included of how to apply the Power Rule even when it isn't immediately obvious.

Thanks again to Rachel for providing her notes for today.

May 4 - Slope of a curve

The main concept for today was the definition of the derivative. Imagine two points on a curve - call them (x, f(x)) and (c, f(c)). Calculate the slope between those points (change in y over change in x) and then take the limit of that expression as x -> c (which implies that the two points "move" ever closer together). The value of the limit can be interpreted as the slope at that point (c, f(c)).

Thanks to Rachel for her notes for today.

Notes from April 30 - Introduction to Limits

Today, we were introduced to limits. At the moment, it's a calculator-heavy concept, but we eventually will learn to do evaluate them without the calculator.

The actual definition of a limit will not be pleasing right now, but for now, we have an "intuitive" definition:

lim f (x) = L

x -> c

where for x values "close" to c, the y values are "close" to L.

April 12 - Proof by Induction

Proof by induction is a tricky little concept and it will take some time to figure it out. It really emphasizes algebra skills (of all things...) and the really disconcerting thing is that it is sometimes hard to recognize you're finished; it may not feel as though you've really done anything.

But.

It is a powerful tool in a mathematician's arsenal and if you continue to take math past calculus, you can most certainly expect to see it again.

Here are my notes from today:

These are the notes from today. We learned to evaluate the sumation of geometric sequences. That equation is on the first page. We also looked at examples of doing this, and I made a note that for the example with the 6th power, there were for complex roots that we DO NOT need to know.

March 31 - Arithmetic Sequences

We defined an arithmetic sequence today in class as a sequence in which there is a constant difference between terms. I gave my version of the formula for the nth term in an arithmetic sequence; note that it differs slightly from the formula in the book.

Thanks to Jaime for today's notes. :)

Sequences, continued

Today, we learned another way to write sequences. In addition to the closed form we learned in class yesterday, we also learned the recursive definition for sequences. The recursive definition allows you to find the next number in the sequence one by one. It consists of two equations: the initial value for a (a₁= some constant) and then the equation for a_n as a function of a_n-1 (a_n= a_n-1 with some operation.)


Also, we learned how to find the sum of a sequence of terms. In the examples in the notes, the "n=" term underneath the Σ tells us the starting value for n. The term to the right of the Σ tells us the operation to preform for each sequential n value. The number above the Σ is a constant, which tells us the final value of n to include.
(In the 1st example, the number above Σ is 5. Since the operation given is n², the final number we include should be 5² = 25.)

Notes 3/29

Today in class we learned about sequences. A sequence is an ordered set of mathematical objects, like numbers. We have seen sequences before, since they are like pattern recognition where you must find the next set of numbers. For instance, the next three numbers in the ordered set 3, 6, 9, 12 are 15, 18, 21. There are different types of sequences, like linear, quadratic, exponential, and factorial. There are examples of each on the posted notes.

Today in class we learned about using DeMoivre's Theorem with roots. We learned that the root affects how many answers you should look for (for instance, square root has 2 answers, and cube root has 3). We also learned that in order to find the other answers after using DeMoivre's Theorem, you have to divide 360 (or 2 pi for radians) by the number of answers that you need and add that number to the theta you had until you have the number of answers you need. After that, the answers will start to repeat.

Notes 3/22

Today we both reviewed material from last week and covered a bit of new material. The stuff we reviewed was converting complex numbers to trig form (which can be done by drawing a triangle to find the r and theta). Next we learned about multiplying and dividing complex numbers in trig form: we use certain formulas (which are found boxed at the bottom of the first notes page and on the second page) and plug in the respective Rs and thetas. We found that when multiplying, the Rs multiply and the thetas add, and when dividing, the Rs divide and the thetas subtract. We also covered complex numbers to a power where we saw that Demoirvre's Theorem could be used.

March 4 Notes: Cardioids and Limaçons

Today in class we learned about the polar graphs of Cardioids and Limaçons.

Polar Graphing of Circles and Roses

We covered graphing of two main different shapes today using polar equations. First, we covered the main equation for graphing a circle, using sin and cos, depending on its orientation. Second, is the main equation for graphing a rose and what information can be taken from the polar equation about the rose. At the top is just a diagram and equation used for the last problem on the yesterday's homework.

In class today, we explored various methods used in converting polar equations to rectangular equations and rectangular equations to polar equations. Our previous notion about the importance of mastering trigonometry identities was reaffirmed, as solutions for problems often involved these identities. Rather than being presented as a new topic all together, the notes served as an additive to yesterday's notes, using similar/the same techniques to serve slightly different purposes.

Polar Graphing (3/1)

Today in class, we learned about polar graphing. The points are written as (r (the directed distance from the origin, θ (the angle drawn in standard position)). An example would be (4, 60°). We also learned how to convert from polar to rectangular by using the formula x=rcosθ and y=rsinθ. For example, you would insert the formula into (4, 180°) to become (4cos180°, 4sin180°) which would solve to (-4, 0). Finally we learned how to convert from rectangular to polar. You would use the formula, x^2 + y^2 = r^2 and y/x = tanθ, and there may be more than one solution.

Parametric Equations Using Trig Functions (2/18)

In class, we discussed conics and their equations in three dimensions. We began using trigonometry to find both the three dimensional and rectangular equations for the conics. We also discussed the relationship between the graphs of the trigometric functions and the resulting graph of the conic when they are used in the equations.

Parametric Equations

In class today we learned about parametric equations. Right now, when given 2 equations we are supposed to graph them on our calculator to figure out the general shape of the graph. Then by using the Brute Force Method, we can find out the rectangular equation for the graph.

Notes for Feb 8

In class we learned about graphing with three variables, involving multiple dimensions.

We learned some of the common graphs you can make with three variables, as well as how to make them. We learned the standard form of these equations as well as how to use these equations to find the velocity of the line, the speed, and the slope.

Parabolas

Today in class we talked about parabolas. Mr. Vischak showed us a picture of a parabola, a hyperbola, and an elipse on his computer. There were two lines on each of these graphs. One connecting from the vertex and one from the focus. We learned another way to write eccentricity. mAB/mAC, with A B and C being the vertex focus and directrix respectively. We also learned how to write parabolic equations. y=a(x-h)^2+k is the standard form.

Hyperbolas- 2/10

Today we discussed hyperbolas, which can be confusing (especially the slope of their asymptotes), but once you memorize their properties you will dominate them.

Ellipses

Today we checked the worksheet and reviewed ellipses.

Notes for 2/3

Today we talked about the dot product of two vectors. Basically, if you have two vectors and want to find the dot product of them, you have to take the first term of each and multiply them together and add that to the second terms multiplied together (the dot product of vector "a" and vector "b" would be a1*b1 + a2*b2). We also learned the equation to find the angle between two vectors, as shown in the notes. By applying what we learned today about dot products and what we learn previously about vectors' magnitudes, we could figure out the angle. We proceeded to do a practice problem with this concept. Something to note about dot products is that if the dot product of any two vectors =0, the vectors are orthogonal (perpendicular in more than two dimensions). Another important thing is the diagram in the upper right-hand corner of the notes.

Notes for 2-2

Today we talked more about vectors especially how to combine them. We worked through several types of practice problems which all required you to be able to combine vectors. For example, one type of problem gives you 2 vectors and ask for a third which, when combined with the other 2, gives you a vector with 0 magnitude. For those, you add the i and j components of your vectors and the final one will have the same value but with the opposite symbol(+/-). Another type of problem we solved asked for the vector that would result when multiple vectors are combined. For those you just add the i and j components and if you need the angle, it helps to draw it out and think of the i component as the x value and the j component as the y value. One thing to note is that these problems are very similar to physics problems that involve vectors of forces.

Notes on 2/1

In class that day we learned about how to find the magnitude of a vector and how to find different components of vectors with magnitudes larger than one. We did a few example problems and we ended with a physics type problem in which a ball is thrown and we use vectors to find its maximum height. We also learned how to use "i" and "j" instead of the standard component form.

Today in class we learned about vectors. We started off class by talking about numbers. We graphed the number 3 on a number line. We found out that you can graph 3 in two ways. The first way is the literal number 3 on a line, so we placed a closed circle on the number 3. The second way we learned was you can use 3 as a value, which is shown on the second number line. We drew closed circles on 2 and 5 and drew a line connecting the two points and drew an arrow showing the direction in which we went. This shows us the difference between a number and a value; one closed circle on the number 3 was the number and 2 closed circles on 2 and 5 was the value 3. We next learned about the Component Form of a vector which goes as follows:

x(with a hat) = (the numbers with the 'a' and 'b' are there for coordinates only)

First off, the "hat" over the x in the picture is there to show that we are dealing with a vector, but on tests vectors will be shown in bold face. Second, we learned that the angled brackets tell us that we are dealing with a vector quantity.

To use the Component Form of a vector, all you have to do is plug in the coordinates for 2 vectors and do the algebra to get your answer.

1/28/09 - Proofs for Law of Sines, Law of Cosines, and more fun stuff...

Today was more of a theoretical class, we worked with 5 problems from the board, 3 proving the Law of Sines, the Law of Cosines, and the Pythargoren thereom, and 2 involving those theorems, some geometry and trig. We started, but didn't finish covering the last two problems, so you can try to figure them out on your own, and wait for the next post to check your work...

one fact we covered in class that was new to me: to inscribe a triangle in a circle, center the circle around the point of intersection of two of the triangle's perpendicular bisectors don't worry about the third, because all three intersect at the same point)

Law of Sines- 1/27/10

In class today, we took notes on the Law of Sines. This formula allows you to solve for the rest of a triangle if you already know an angle, angle, and a side (AAS), or if you know a side, side, and an angle (SSA). This formula allows you to solve for the unknowns of all triangles as long as you have enough given information. When you have an AAS triangle, there is one possible solution, but with SSA triangles, you can have 0, 1, or even 2 possible solutions, so it is important to check if triangle even exists.

The basic formula for the Law of Sines is:

 (a/sinA) = (b/sinB) = (c/sinC) 

Notes 1/26 - Law of cosines

1/26 - And we're back to school. . . Today we took notes on "the law of cosines." This formula allows you to solve for any triangle (find all values for angles and sides) if at least 3 of these values are known. The law of cosines works with any type of triangle - not just right triangles. One thing to remember in notation is that uppercase letters refer to angles and lowercase to sides. In solving for a triangle using the law of cosines it does not matter which value you look for first, as long as you can find the others taking that approach. Also, remember to check if the triangle is even possible in the first place. The following is the formula for law of cosines:

a^2 = b^2 + c^2 - 2bc*cosA