AP Calculus BC Review April 7th

Today I reviewed the Intermediate Value Theorem. The Intermediate Value Theorem basically states that as long as a function is continuous over a given interval [a,b], there is a value of that function (x) that lies within that interval. The function must be continuous because it would have holes, gaps, jumps, etc. Lastly, the IVT states that a number exists but we may or may not be aware of that number.  

AP Calculus BC Review April 6th

Today I reviewed how to calculate volumes of solids. You use different methods depending on the situation/problem that you are given (disk, washer, and shell methods).

Disk:
*You use this method when a solid is created by rotating an area around a given line.
Formula- Integral from a to b of (pi * r^2)dr.

Washer:
*You use this method when trying to calculate the volume of a solid with a known cross-section.
Formula- v = (the integral from a to b) of A(x)dx.

Shell:
*You use this method when you revolve the object around the y-axis.
Formula- v = 2 (pi) * (the integral from a to b) of p(x)h(x)dx.
*p = the distance from the center of the rectangle to the axis of revolution.
*h = the height of the rectangle.  

AP Calculus BC Review April 5th

Today I studied Euler's Method. I decided to see a Khan Academy video regarding Euler's Method in order to freshen up on the key concepts of it (https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/eulers-method-tutorial/v/eulers-method). This method is the simplest of numerical integration. At the end of my study session, I realized that three very important concepts are needed to apply Euler's Method.

1) A starting point (x.y)

2) The change in "x"

3) The slope (of each line segment)

By using the chart below, we are able to easily apply this method:

(x,y)	Δx or dx	dydx	dx(dydx)=dy	(x+dx,y+dy)

*Lastly, a very important thing I discovered is that Euler's Method is basically all repetition. You just need to keep on repeating steps in order to discover the desired "x-value."

April 4th AP Calculus Review Session

Today I studied basic integral functions. I reviewed the basic integral rules that you add 1  to the numerator and for the denominator copy the exponent and add 1 as well. I also reviewed the integral rules for sine and cosine as well as for the natural log. When there is a definite integral, the integral is bounded from a to b. Another key concept to remember is adding the +C at the end of integrating. You do not need to do this, however, when it is a definite integral.

AP Calculus BC Review April 3rd

Today, after realizing that I forgot to study yesterday, I decided to quickly brush up on a recent topic that we've learned, parametric equations. I watched a brief video regarding parametric equations (https://www.youtube.com/watch?v=eun8uu3k6Ig) and reviewed that with parametric equations, there are two functions (x and y) and that these functions are functions of time. The video talked about how to calculate the slope of a tangent line, velocity vectors, accelerations vectors, and speed. To finish up my studying, I then completed the 2004 (Form B) Practice Question 1.

Assignment #15

1) 0^0 = 1 because anything raised to the 0 is one (x^0=1).

2)

a) T5(x) = (x^2/2) + (x^3/6) + (x^4/24) + (x^5/120)

b) T5(x) = (x^2) - (x^3/6) + (x^5/120)

c) T5(x) = (-x^2/2) + (x^4/24)

Assignment #14

The man in this paradox will never catch up to the tortoise, let alone overtake it, because as the man gets to the tortoise's initial position, the tortoise would have moved forward more. He will never actually catch up to the tortoise because of this. This is similar to Zeno's paradox because in this paradox, the man will never actually reach the wall because he keeps on taking steps that are half the size each time. There will always be a gap between him and the well. After an infinite amount of time however, the man will catch up to the tortoise and overtake it and the other man will eventually reach the wall in real life. This is because the ratio is 1/2 and it is convergent.

Moreover, I agree with the answer of .5 because .5 is between 0 and 1. It is impossible to know what the correct answer is. Also, this relates with what we have been studying as it relates to the alternating series. I believe that the reasoning behind Thomson's lamp dilemma makes sense because there can be multiple answers.