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Parabolae in the physical world

In today's world, mathematics and science play a very vital role in re-shapping the way our society view the world today. In terms of the facilities that are around us, mathematicians and scientists have contributed alot in order to improve our living standards. Natural science and math leave their marks on everything around us, everything we see and utilize have its own math and science behind them. From aircrafts to automobiles, from bullets to CERN, mathematics and science have revolutionize the world that is around us. For example; Parabolic motion is imperative for physicist in order to calculate the velocity for the "x" and "y" components of a ball being projected from position "z". Of course there are other factors such as wind resistance and drag that would undermine the legitimacy of the use of parabolic motion in this situation. Nevertheless, every complex calculations have to start somewhere, and one of the key way mathematicians and physicist come up with a hypothesis for the velocity is via a simpler assumption and calculation, dismissing drag, wind resistance and other factors; the use of Parabolae motion. By using the value for the max height, displacement of the ball, mathematicians are able to generate a quadratice formula in order to come up with a hypothesis or generalization regarding the overall motion of the ball. The example below shows the trace of a ball being thrown from point "x" to point "y".

[1]

The example above shows the trace of a ball being thrown from point "x" to point "y". From the data gathered from the traces of the ball, and time taken for the ball to go from point "x" to point "y", mathematicians would be able to model the overall motion of the ball by using a quadratic formula; Ax^2 + bx + c. This way, investigations would be able to be conducted to calculate the max height of the ball, in mathematical term, to evaluate the value for the vertex and the axis of symmetry. Reiterating the use of parabolae in the to calculate the velocity for "x" and "y" component of the projectile, the example below demonstrate how it is applied in the physical world;

[2]

To conclude, every object that is thrown up at any direction, would follow a parabolic motion due to gravity, hence any mass that is thrown at a projectile follows a parabolic motion; that can be expressed via using the standard format of a quadratic equation; Ax^2 + bx + c.

Application of Calculus in Economics

Calculus play a very big role in the field of economics. Since it is a discipline that involves the study of distribution of resources economists have to show an excellent skill in Mathematics as they hold the responsibility to calculate matters such as; the required input, expected output, labor and machinery resources available to produce the end products. As an economist, one is expected to be able to generate a hypothesis or future predictions based on mathematical calculations. One way economist does this is by plotting the data that are given to them on a set of "x" and "y" axis, from there, a mathematical model is derived in order to analyze the economic trends. Without calculus, calculations involving any factors that are associate with rate of change is impossible.

In Economics, the cost function is labelled as "C(x)". As the economy fluctuate, the market price of a particular product can possibly increase or decrease. If there is a shortage in resources, the cost to manufacture the product can increase by a factor of "h", making the price difference to be calculated as C(x+h)-C(x). However what economists are interested in is usually the average cost of producing a specific product at certain periods. To compute this, [C(x+h)-C(x)] / h is the equation that is used. As shown, this equation is similar to that of the definition of a "limit" in mathematics which is; f'(x)=[f(x+h)-f(x)]/h. Any values computed by the C'(x) function is known as "Marginal Cost". The following is an example taken out from an Mathematic textbook regarding the application of the cost function:

[3]
As shown above derivative is utilized in this problem in order to find the not only the "marginal cost" function, but to calculate the estimated cost of producing one more unit based on the given Cost Model. In this given example however, instead of using the definition of limit, a different approach is used in order to solve the problem. Regardless of this, this example clearly highlights the importance of calculus within the field of Economics. This example is only a small portion of the application of calculus in economics. There are many more examples that require the knowledge of calculus. Of course, economists would always be interested in finding ways of maximizing profits and minimizing cost. For efficiency to be maximized, the "marginal cost" would have to be equal to "average cost". Average cost per item is written as A(x)=C(x)/x given that "x" is the number per item produced. Therefore, average cost per item is at its lowest value when C'(x)=A(x). Refer to the following example:
[4]

Without calculus, this calculation would have never been possible to compute. In conclusion if there is one thing economists are interested in, it is "rate of change", and with the amount of trends they analyze from graphs, avoiding the use of calculus is completely inevitable in this field. Rate of increase production, rate of demand, rate of decline, Calculus is certainly a tool economists apply in their field of work in order for them to make reasonable hypothesis regarding the future of the global economy we live in today.

References

Extracts / photos taken from the International Baccalaureate Higher Level Physics study guide for the 2007 edition

[1]Kirk, Tim. "HL - Projectile Motion." Physics. New York: Oxford UP, 2007. 80. Print.

[2]Kirk, Tim. "HL - Projectile Motion." Physics. New York: Oxford UP, 2007. 80. Print.

[3] Owen, John. "Mathematics for the International Student." Mathematics SL. Australia: Haese & Harris Publications, 2004. Print.

[4] Owen, John. "Mathematics for the International Student." Mathematics SL. Australia: Haese & Harris Publications, 2004. Print.