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.