Review 4/6

Today I reviewed Riemann sums. In order to help me review I watched a Khan Academy video.
https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/riemann-sums/v/simple-riemann-approximation-using-rectangles
It explains how the Riemann sums are used to approximate the area under a curve. Because it is an approximation it does not give us an exact value, but sort of an idea of one. You break the curve up into rectangles and solve for the area of each one and add them up.

Review 4/5

Today I reviewed antiderivatives and indefinite integrals. To help me review I watched a Khan Academy video. https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/indefinite_integrals/v/antiderivatives-and-indefinite-integrals
In this video it explains the relationship between derivatives and antiderivatives. When taking the derivative of a function any constant value becomes a 0, and any x value is multiplied by its exponent and also subtracting one from the original exponent. Some examples are
x^2 + 1 ----> 2x
 x^4 + x^3 + 3 ------> 4x^3 + 3x^2

When taking the antiderivative, also known as the indefinite integral, you take the coefficient of the x value and make it the exponent. For any constant you multiply it by x. Finally you must add a +C to the function. You must do this because there could be a value whether it is 0, 1, 10 ....
Some examples are

2x ---> x^2 +C
3x^2 +5x -4 -----> x^3 + (5x^2)/2 -4x +C

Review 4/3

Today I decided to review parametric equations. To help me while I reviewed my notes I additionally watched a Khan Academy video helping explain what parametric equations actually are. 

https://www.khanacademy.org/math/precalculus/parametric_equations/parametric/v/parametric-equations-1

In the video it explains that both x and y are functions of time. Through the use of parametric equations we can determine the direction and path of an object. Additionally you can also determine a third parameter, Z, on a three-dimensional plane. You can also determine the slope of a tangent line, velocity functions, acceleration functions, and speed. 

I also google searched problems with their solutions to help me understand and practice. http://tutorial.math.lamar.edu/problems/calcii/parametriceqn.aspx 

Assignment #15

1) Anything raised to 0 is equal to one, therefore 0^0 = 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

1- The man will never be able to catch up to the tortoise. As he continues to chase the tortoise to its initial position, it continues to move forward creating a new position. Because of this he will never be able to pass the tortoise. This is similar to Zeno's Paradox because in this paradox the man can never reach the wall. The man continues to half each of his steps. Since he continues to half each of his steps, he will never be able to reach the wall. If there is an infinite amount of time, the man will eventually catch up and over take the tortoise. Additionally the other man will eventually reach the wall.

2- I agree with the answer of .5. .5 is between 0 and 1. This gives an approximate idea of what the answer could be but it is impossible to know the exact answer. This is also related to what we have been learning because it relates to the alternating series. Thomsons lamp dilemma makes sense because there can be an infinite amount of answers.

Assignment #13

In order to find the volume of the solid when revolving f(x)=1/x around the x-axis you must do V=integral of (1/x)dx*π  [0,∞) V=(ln∞-ln(1))*π V=π

In order to find the surface area you do
Surface Area=((1-(x^-4)^1/2))dx from [1,∞)*π*ʃ1/(x^2)dx = ∞

This isn't a paradox because the volume of the solid approaches π as the functions continues unto ∞ and while the surface area does not approach a definite integer while the function continues to approach ∞.

Assignment #12

This what if is about the demographics of Fairies. It relates to the logistic curve with both human and fairy populations and how they increase or decrease in the environment. As the human population increases so does the fairy population. The fairies are also immortal, meaning they cannot die unless something learns how to kill them. As the human population increases the carrying capacity, eventually the population would decrease slightly and since there is a direct relation to the fairy population, but they are immortal, the fairy population levels off and stays at a specific population.