【微分積分】九州大学2022年度入試理系数学問５の解説と解答例(第113回)

[Differential and Integral Calculus] Explanation of Question 5 of the 2022 Kyushu University Entrance Examination, Science Stream Mathematics. The Kyushu University exam questions this year seem generally difficult. As the first in a series, I will explain Question 5, which uses parametric equations. There are four sub-questions, but the guidance leading to Question (4) is excellent. Question (1) simply requires differentiation, but it will be quite difficult if you don't remember the sum-to-product formula. If you remember it and pay attention to the domain, it can be considered relatively easy within this problem. However, don't overlook the fact that it's dy/dx. Question (2) asks to "find the area," so you can imagine that integration is necessary. However, the result of Question (1) is key to finding the integration interval. You must realize this. Also, when you actually start integrating, you need to know variable transformations, and furthermore, the half-angle formula and the product-to-sum formula are also necessary. Question (3) is about symmetry and curl. For symmetry, it's helpful to recall the proof of even functions. Also, when it comes to rotations, it's easiest to think of complex numbers. It's best to consider the xy-plane as the complex plane. However, this will involve a considerable amount of calculation. A while ago, we would have learned about "matrices," so using those would be an option, but you'll learn about that in university. With that in mind, question (4) awaits. You'll need to calmly organize what you know and draw a diagram. This is quite a challenging problem. Why not try it as a test of your skills? Conversely, if you don't experience it at least once, you won't be able to solve similar problems when they appear, so even if you're not confident in your abilities, please give it a try.