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.    

Assignment #13

In order to find the volume (V) of the solid when revolving "f(x)=1/x" about the x-axis you must do the following:

V=integral of (1/x)dx * pi [0,infinite)

V=(ln infinite - ln(1)) * pi 

V=pi

In order to find the Surface Area (SA) you must do the following:

SA=((1-(x^(-4)))^(1/2))dx from [1,infinite) * pi * integral of 1/(x^2)dx

SA=infinite

This is not a paradox because the volume approaches a finite number (pi) as the function approaches infinite and the surface area does not approach a finite number as the function approaches infinite.

Assignment #12

This What If talks about Fairy Demographics. It relates to the logistic curve as it talks about both the human and fairy populations and how they increase/decrease in the environment. Fairy demographics differ a bit as fairies are immortal meaning they do not simply vanish (unless something learns to kill them off). The fairy population keeps growing as the human population keeps growing because both correlate. As humans reach the environmental carrying capacity of 9 billion however, it will be difficult for them to survive and the human population will decrease. The fairy population, on the other hand, would level off (it will not increase or decrease) as fairies are immortal.