Calculus Investigation

This task is aimed to investigate the development of mathematics before and after calculus, and to make possible conjecture of how it will develop further in the future. Calculus is a method of finding the gradient, a measure of how quickly the function is changing as the x-coordinate changes, of a tangent to a curve. The maximum or minimum points, which are the turning points in a graph can also be deduced using calculus. It can also be used to determine the optimal design of different objects – one that uses the smallest amount of material to hold a given capacity. Other examples include that biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food

Questions unable to solve in the past became solvable due to the discovery of calculus. The purpose of this task is to see how significant calculus has changed the method of calculating the same thing before and after its discovery, and from this, to make possible assumption of future development of mathematics. My planning is to first find out an application of calculus at present times, and research how people in the times before the discovery of calculus calculated the same thing. Then, to do the calculation with calculus and compare it with the previous calculations. In this stage, calculation skills and comparative thinking will be used. The results will then lead to a conclusion of how mathematics has developed and the assumption of how mathematics will develop in the future. In the stage of investigating how people calculate the same thing before and after calculus, research skills will be used to find out information. From this, application and calculation skills will be used to demonstrate the methods of

It is chosen for this task because it has a shape much closer to a regular cylinder and thus the errors could be minimized if it was used. If the shape is not regular, which means it is not in the shape of regular geometry, a method developed by Archimedes which is very famous in the world can be used to calculate the volume of it. It is to put the object into a known amount of water, then calculate the amount of change in the volume of water. The difference is then the volume of the object. However, this method does not tell whether the volume is optimal or not, therefore, calculus will be conducted in the following part to find out.

The measurements are listed as below: diameter: 5 cm (radius: 2.5 cm) height: 9.5 cm

The volume written on the can is 190ml, therefore, the aim of my calculation is to find out, using calculus, the optimal volume of this cylinder from the measured surface area and compare the result with the volume written on the can.

First, calculation before the discovery of calculus will be demonstrated.

In Kepler’s calculation, he used a more geometrical way to see the