Mathematics HL Extended Essay (Grade A)

International Baccalaureate Extended Essay

Subject: Mathematics

Research Question: To what extent can we approximate the antiderivative of functions that are non-integrable or have non- elementary antiderivatives, without using a calculator?

1. Introduction

When I was doing my Mathematics Internal Assessment, which was on comparing
the efficiency of bottles according to their shape, I remarked upon, the high reliance on scientific calculators when doing such a project. In order to find the surface area of each bottle, I had to use Calculus which involved integration of complicated functions. Since calculators provided me with results of complicated calculations which consisted of lengthy decimal numbers, I strongly felt it would have been impossible without the use of a calculator. This made me ask myself whether everything in the field of Mathematics is possible using this electronic device which performs such mathematical calculations for us. I thought that there must be some mathematical areas where calculators cannot help us, as there are some sectors where machines and robots cannot replace human labour. For example, despite an escalating scientific revolution throughout the last few decades, surgery robots cannot be implemented in psychiatric hospitals where doctors need to deal with patients wholeheartedly with emotion and giving them an advice. Likewise, I thought that even though it is clear that calculators make our lives easier, but it should have limitations as well.

While thinking of problems which calculators cannot solve, it reminded me of the case where an online calculator could not perfectly integrate the question my math teacher gave us. This made me ask myself whether it would be possible to integrate that function which the calculator could not. And if so, to what extent will it be possible to approximate the value of the indefinite integral of functions without a clear antiderivative, without the use of a calculator…