La Grange and Alternating Series Error- Tuesday, April 7

Today we learned about how to find an alternating series error and the La Grange error bound. Both of which are used to approximate a value of desired accuracy. Both methods require the use of the next term in the series to be used and compared to the one before it. The difference is that the alternating series error can only be used when there is an alternating series. The La Grange error bound requires a Taylor series in order to be used.

Notes 03-30-09

Today we continued with Power Series and focused on two particular types: Taylor Series and Maclaurin Series. We first defined the two sub-series, then particed constructing them given different conditions.

Power Series

Today in class, we learned about power series and how to write them. Some power series are centered at x=0. These series are defined as the infinite sum of A sub n (An) multiplied by x^n. Other series are centered at x=c and these are defined as the infinite sum of the An multiplied by (x-c)^n. When writing a power series we first make sure the given function is in the geometric series form of 1st term/ (1-ratio). Then, the rest is easy...you just write the first term in the series and multiply each of the following terms by the ratio. Remember, the absolute value of the ratio must be less than one and to find the values of the interval of convergence for the power series just solve that absolute value equation.

Pythagorean Tree

This is the video we recorded in class today showing the Pythagorean Tree fractal.

(Note that clicking on the link will take you away from the blog...)

Notes from Class March 23-7th Period

Today we learned the Ratio Test. We were able to determine convergence or divergence of a series by dividing its sequence of a(n+1) over a(n). We also derived e by using the Ratio Test. Notice how the Ratio Test is similar to the Geometric Series Rule (r in geometric series is term/previous term).

Idea of Partial Sums

I'm posting a video of the partial sums graphs that we looked at in class before spring break. This particular post may or may not prove useful, but I am trying out the idea of making and posting a video.

Partial sums video

(still working on embedding...)

Comparison Tests

Today in class (Wednesday 3/11), we learned about how to determine if an alternating series converges (conditionally or absolutely) or not. An alternating series is most easily recognized by (-1)^n involved somewhere, which will alternately change the values of the sequence from positive to negative. We learned some cool stuff about what it means for a series to converge as well, and discovered that certain sums can be different values depending on the order of the terms.

On March 9th in class, we discussed a new way to see if summations are converging or diverging. We now know that if the definite integral from 1 to infinity converges, then so does the respective summation. We also learned the P-Series Test, which is very specific in its application; it can only be used for functions that are 1/n^p.

In class March 5th we discussed sequences. We talked about both geometric and arithmetic equations and how to solve them. We also discussed combinations and permutations. And, we mentioned convergence and divergence and how to determine whether something converges or diverges.

Arc Length of a Polar Curve

Today we learned how to derive and find the arc length of a polar curve.
We also learned how the arc length formula of a parametric function can be applied to find the arc length of a rectangular curve.

Parametric Equations-Arc Length (Feb 18)

Finding the length of the segment between two points on the curve is the most accurate way we can use to find the arc length between the two points. We can then relate this to the Pythagorean Theorem, and set up a definite integral using this idea to find the arc length. SORRY THIS IS SO LATE!!!

Parametric Motion 2/19/09

Today in class we discussed Parametric Motion dealing with finding position, velocity, speed, and/or distance at a certain time or time interval from a parametric equation.

2/23/09 Notes

Today in class we briefly discussed/reviewed translations in polar equations. We also learned about differentiating polar equations and how to apply it.

Notes Feb 20

Polar equations. Today we talked about graphing polar equations. We went over the different shapes to expect along with converting to rect. eqn's.

2/17/09 Notes

Today we took second derivatives of parametric equations and used them to find concavity.

2/16/09 notes

Today we covered parametric equations, graphing them and solving them. We also talked about velocity and speed and the two basic parametric things we should be able to recognize. In class, we did a problem on finding the slope of a tangent line of the equation.

Missing Days

I just wanted to put up a post to note what happened between Michael's previous post and Sabrina's post to be...

On Tuesday, Feb 10, we reviewed differential equations and looked at some application problems.

On Wednesday, Feb 11, there was a test over differential equations, improper integrals, some new techniques of integration (including integration by parts and by partial fractions), slope fields and Euler's method. The test average was 90.1.

On Thursday, Feb 12 we hunted space monsters.

There was no school on Friday, Feb 13, and that brings us to Monday the 16th.

Improper Integrals

Today we learned how to solve improper integrals, which are definite integrals that are based off of limits. The most important concept was that improper integrals that equal infinity have diverging graphs, and those that equal a rational number have converging graphs.

Class Notes, Friday the Sixth Oh-Nine

Friday's topics involved solving integrals using calculus. Our main method was integration by partial fractions - using algebra to break up, via factoring, an otherwise un-integralable denominator. Other forms of attack included breaking a fraction into smaller parts by distributing the numerator and completing the square.

Notes February 5

Today we continued working with u and dv, but it got a bit trickier. We completed two problems, which took a while to complete and had many steps.

Integration by Parts

Today we learned integration by parts. We can use this when integration by substitution (the one using u) doesn't work.

Today we just used the logistic model from yesterday in class, generated a slope value and found out what the logistic model would look like if we started with different values.

January 29

Theoretically, Euler's Method approximates a curve by using line segments. The smaller the delta x, the shorter the segments, and thus usually the more accurate the approximation.

You do need to know how to do Euler's Method by hand, but it can get tedious so it is appropriate to use the calculator program (regularly). Don't forget to put the differential equation in Y1.

for Peter (and 4th period...)

Peter wanted to know how to do something so I'm typing...

.....

This seems like it should work well.

January 28

I introduced slope fields today. Here are my notes for class.



This is what a slope field looks like when generated on a computer:




There really isn't going to be anything complicated associated with this topic. There's this thing called Euler's Method that we'll do tomorrow, but it is really just applied arithmetic. Really.

Tuesday, January 27

Here are notes from class today. Mostly a revisiting of solving differential equations.



The algebra step of going from ln y = x + C to y = Ce^x is a key concept.

a communist society...

So this is not _my_ blog, this is _our_ blog.

I guess I'm in charge, but what I find appealing about this setup is that all of you are enabled to be authors and so can add your thoughts at any time. I think that ultimately the structure here will be exactly what I envisioned, both in form and creation.

This is new territory for me though, so I expect it will evolve as I (we) gain in experience and wisdom.