AP Calculus BC Review April 12th

Today I studied the Logistics Equation. These equations describe growth over time and often deal with factors associated with populations because populations grow over time. To solve the dilemma associated with infinite growth (which can't exist in realistic situations), logistics equations implement a carrying capacity, cutting off this unending growth. Lastly, to solve logistic equations, it is very important to separate the variables (x and y) so that we can be able to integrate. If we do not do this, we would not know what we are solving for.

AP Calculus BC Review April 11th

Today I decided to review Trapezoidal Approximations of Area under a curve. I saw a khan academy video to familiarize myself with the topic (https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/riemann-sums/v/trapezoidal-approximation-of-area-under-curve) and then practiced a few problems on my own. This rule is used to calculate the definite integral. You approximate the region under the function f(x) and calculate the area as a trapezoid. The formula is this:



I find that these problems are quite easy and repetitive as long as you follow the fornula.

AP Calculus BC April 10th Review

Today I reviewed the Fundamental Theorem of Calculus. I began by watching a khan academy video to review the basic principles of the theorem (https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/fundamental-theorem-of-calculus/v/fundamental-theorem-of-calculus). The theorem basically links the concept of a derivative of a function with the function's integral. The first fundamental theorem states that the definite integral of a function is related to its antiderivative. The second fundamental theorem states that the definite integral can be sought out with an infinite number of antiderivatives.

AP Calculus BC Review April 9th

Today I reviewed the velocity, acceleration, and position functions. The position function's derivative is the velocity function and the second derivative of the position function is the acceleration function. This observation is important because it shows us the relationship between these functions. It is also very important when dealing with problems asociated with real-life scenarios and physics-type questions ('Degrees kids' that's you guys... Always use 'Radians Mode' though...). To finish off my studying I did some problems from Sample Exam III.

AP Calculus BC Review April 8th

Today I studied the first and second derivative tests. The first derivative test is used to find the intervals in which the function is increasing/decreasing. The second derivative test is then used to calculate a curve's concavity and the values of any relative extrema. Lastly, the point of inflection determines where a graph changes concavity. To find this point, you must set the second derivative equal to "0" and solve.  

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.    

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.

Assignment #11

1) The lecture was very informative but also moved a bit quickly. A lot of information was presented in a short amount of time and students had (it seemed) to be constantly aware of what was going on. Not paying attention for even a few moments can be dangerous. The class size was also very large. I expect a lecture in college to be very similar to this.

2) Courses I would Take:
- Biological Engineering
- Civil and Environmental Engineering
- Economics
- Electrical Engineering and Computer Science
- Global Studies and Languages
- Mathematics
- Mechanical Engineering
- Physics

There is some value to a university even if lecture videos are online because attending a lecture in person truly gives you the "university experience." You can physically interact with others, ask questions, and make study groups if you need assistance. Watching the lecture video should really only be used after attending the lecture itself in order to reinforce what you just learned.

Assignment #10

1) Logistic equations represent growth over time which is perfect for describing populations because populations can grow over time as well. Environments cannot continue to grow forever however and logistic equations solve this problem by implementing a carrying capacity.

2) The point is when the limited is divided in half (L/2). This point represents the point of maximum growth of the logistic curve.

3) The first step is to always separate the different variables (usually x and y) so that we can integrate the equation. If we do not separate the variables then we do not know what we are solving for and the whole understanding of the problem becomes null.

Assignment #9

The video discusses the splitting and averaging of surfaces in order to create the animations we see in Pixar movies. Pascal’s Triangle is used in order to create smooth curves and shapes (for the limits) for various objects. For surfaces however, Pascal’s Triangle does not work and a different set of mathematical tools must be used in order to figure out weights that will generate smooth objects. Tony (the non-British man that is talking) discusses approaching infinite in order to create these smooth shapes. By splitting and averaging shapes an infinite number of times, Tony is saying that the two points will get closer until they reach a specific limit and come together at the shape’s original midpoint. Weights are carefully chosen in order to produce the smoothest surfaces necessary.

Assignment #8

1)

A) ʃsin u du = -cos u + C
B) ʃcos u du = sin u +C
C) ʃtan u du = -ln |cos u| + C
D) ʃcot u du = ln |sin u| + C
E) ʃsec u du = ln |sec u + tan u| +C
F) ʃcsc u du = -ln |csc u + cot u| + C

2)

You would set "u" equal to "2x" because it is inside the function "u^1/2". du would equal 2dx but it cannot be achieved because of the (4x+1)dx.

Assignment #7

1) The general solution to (x^n dx) is [ (x^(n+1)) / (n+1) ] + C. The C is very important because it represents a constant that will differentiate one integral from another.

2)
sin(x) dx = -cos(x) - It is going in reverse (up instead of down) of S,C,-S,-C
cos(x) dx = sin(x) - It is going in reverse (up instead of down) of S,C,-S,-C
sec^2(x) dx = tan(x) - Memorization, since secant is squared I think of tan
sec(x)tan(x) dx = sec(x) - Memorization, since there is a sec and a tan I think of sec
csc^2(x) dx = -cot(x) - Memorization, since co secant is squared I think of a negative co tangent
Integral csc(x)cot(x) dx = -csc(x) - Memorization, since there is a co secant and a co tangent i think of a negative co secant

Assignment #6

The Second Derivative test is used to determine at what x-values a differentiable function can have relative extrema by seeing if the critical value plugged into it makes the equation positive or negative. If the critical number makes the function a negative the function would be concave down with a relative maximum. If the critical number makes the function a positive the function is concave up with a relative minimum. The First Derivative test is used to find these critical values that can be used in the Second Derivative test to determine the function's concavity, relative minimum, and relative maximum.