User:AdamsNguyen

Hi There!, my name is Adam Nguyen and I am a first year Arts Student at UBC. --AdamsNguyen 02:12, 18 September 2010 (UTC)

Essay Assignments

Calculus in Conservation

Caluclus is simply defined as “the study of change”. Understanding this, one can see its vast practicability and application in all fields of careers. This “change” that one might study could be a change of price with respect to demand or the change of temperature with respect to green house gases. The possibilities are endless.

One particular field of interest is the environment and its conservation and sustainability. Here again, calculus is pivotal in the advancement and understanding of the environment. When considering the sustainability and the conservation for a particular ecosystem, one must be able to understand the dynamics of the organisms living within it. Most importantly, one must understand how much organisms, or more specifically, how many individuals of a given species that the ecosystem can sustain. This population limit, influenced by factors such as resource and space, is also known as the carrying capacity. Once understanding the carrying capacity of a particular environment, scientists and conservationists can effectively construct a plan for its future with minimal effects and consequences to its sustainability.

Furthermore, we can even predict the growth rate of the population, the change in number of individuals in the population per unit time. If we can determine the carrying capacity of a habitat, the instantaneous growth rate of specie and its initial population, we can derive its growth rate. A logistic growth equation can be formulated with all these parameters to determine a growth rate of population:

In this expression,
dN/dt = growth rate
r = instantaneous growth rate or per-capita rate of increase
N = the initial population
t = time (year)
K = Carrying capacity

(K-N)/K = the proportion of individuals that can be sustained within the carrying capacity By dividing (K-N) by K, we put the individuals into a proportion that can be sustained in the environment. The carrying capacity is influenced by many factors including food, water, soil quality, space, breeding sites, predation and diseases. As we can see in this equation, if N is a small number with respect to K, the change in population per unit time, or growth rate, is relatively high. However, if N is close to the K, then (K-N)/K will be close to 0 and the growth rate will be of a lower number.

Here is an example of the logistic growth equation in action:

We have a population 40 bonobo chimps in a habitat that has a carrying capacity of 450 bonobo chimps and the instantaneous growth rate of these chimps is 2.5. What is the growth rate of this population? We will use the logistic growth equation and our following function will be:

dN/dt = (2.5)(40)((450-40)/450)
dN/dt = 91.1111111

The change in the number of individuals in this population per unit time (year) is 91. This means that we would expect to see this population of bonobos to grow to 131 by the following year. This kind of key information can be very important to a researcher who wishes to rehabilitate a damaged habitat.
As you can see with this example in the theme of conservation, calculus plays a very important role in all kinds of careers and jobs.

The Pythagorean Theorem

The Pythagorean Theorem can be simply explained through the equation: $a^{2}+b^{2}=c^{2}$. This theory states that the squared value of the hypotenuse (the long side of a triangle) in a right triangle is equal to the sum of the squared values of the two other sides in the right triangle. With this useful knowledge, we would be able to solve many things relating to triangles! Some people would only think that the application of this theory is only contained within a classroom, but they are wrong! The application of the Pythagorean Theorem is also extensively used in real life situations as well. Here is a simple situation where one would be able to use the Pythagorean theory to resolve his or her problem. Imagine that you are at the south-western corner of a rectangular field that stretches 5 km long and 3 km wide. You have a friend that is on the north-eastern corner of this rectangular field and you want to meet her. Would it be closer to travel along the sides of this field or to travel straight through the field towards her? We can use the Pythagorean Theorem to resolve our problem! We can diagonally split this field so that we have a right triangle. The distance (d) between us and our friend on the other corner can be expressed through the equation $d^{2}+3^{2}=5^{2}$ We then use simple algebraic techniques to simplify this equation and we find that the distance (d) is only 4 km! That is twice as short of a distance compared to walking along the edge of the field. Although this situation is relatively simple, if we can look at everyday life with the perspective of triangles, then the application of the Pythagorean Theorem in real life is limitless!

Mueller, Guntram, and Ronald I. Brent. Just-in-time: Algebra and Trigonometry for Calculus. Boston: Pearson/Addison-Wesley, 2005. Print.

Katz, Victor J. A History of Mathematics. New York, New York: Harper Collin, 1993.