Summary

The web content presents a calculus challenge from the Oxford MAT 2021, focusing on finding the area between two curves using principles of symmetry.

Abstract

The article discusses a calculus problem from Oxford University's Mathematics Admissions Test (MAT), which involves intersecting curves and the application of reflectional symmetry to compute the area between them. The challenge is to find the intersection point of the equations ( y = e^x ) and ( y = 1 - e^x ), and then calculate the area of the region bounded by these curves. The solution provided utilizes the symmetry of the region to simplify the calculation, requiring only half the area to be computed and then doubled. The article encourages readers to engage with the problem by pausing to solve it themselves before revealing the detailed solution process, which includes graphical representations and integral calculations.

Opinions

The author believes that symmetry can significantly simplify complex calculus problems.
Engaging with the problem by attempting it before viewing the solution is highly recommended by the author.
The author finds the use of emojis to hint at the concept of symmetry in the problem both playful and instructive.
The article implies that the Oxford MAT is a prestigious exam, indicative of the caliber of problems encountered by students aspiring to study mathematics at Oxford University.
The author values reader interaction and invites comments to discuss different approaches to solving the problem.
There is an appreciation expressed for the aesthetic aspect of mathematics, as seen in the comparison of the symmetrical region to a butterfly.
The author encourages readers to support their work by sharing the article, clapping for it, or providing monetary support through a "buy me a coffee" link.

Oxford MAT

A Calculus Challenge from Oxford University

Symmetry is your friend 🦋

Oxford university is one of the best places to study at. Every year, thousands of students take the MAT in hopes of studying mathematics degrees there.

Today’s puzzle is a calculus challenge from Oxford MAT 2021.

Here’s a hint: 👜💎🐽🐥 all these emojis are symmetrical in some way …

I recommend you pause the article, grab your pen and paper, and give this a go. When you are ready, keep reading for the solution! ✒️

Solution

Let us look at the graph again.

Our job now is to find the intersection of the given equations, which are the blue and red curves as shown.

The intersection point is at

Let us substitute that into y = e^x see what we get.

Do you notice something about this value? In fact, the horizontal line y = 1/2 bisects the region we are looking to compute!

The point of intersection we have previously found is (-ln(2), 1/2)

Reflectional Symmetry 🦋

Like this butterfly, thanks to the reflectional symmetry of the region, we can compute half the area of the region and multiply it by 2.

The blue region can be found by orange rectangle from the integral under y = e^x: 💙 = integral -🧡

And since that’s only half of the area, we multiply by 2 to get the area of the region.

Mathematically, that is

And that’s our answer.

How amazing 👧

What was your thought process this time? Comment down below, I am eager to know :) 💞

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