First of all, from the content point of view, preparatory mathematics usually covers most of the high school mathematics, but pays more attention to the combination of theory and application. For example, preparatory mathematics may explore topics such as function, calculus and linear algebra in more depth, rather than simply applying these concepts to solve specific problems. In addition, preparatory mathematics may also include some more advanced branches of mathematics, such as probability theory, statistics, discrete mathematics and so on.
Secondly, from the difficulty point of view, preparatory mathematics is usually more difficult than high school mathematics. This is because the goal of preparatory mathematics is to provide students with a college-level mathematical foundation, so its content and requirements are more stringent. For example, preparatory mathematics may require students to understand and master some complex mathematical concepts and theorems, such as Taylor series and Lagrange multiplier method.
Finally, from the perspective of teaching methods, preparatory mathematics usually adopts more rigorous and systematic teaching methods. This includes using more abstract and formal language to describe mathematical concepts and using more complex and in-depth proof methods to verify mathematical theorems. In addition, preparatory mathematics may also emphasize the cultivation of students' independent thinking ability and problem-solving ability, rather than just memorizing and applying formulas.
Generally speaking, the main differences between preparatory mathematics and high school mathematics lie in the depth and breadth of content, the improvement of difficulty and the change of teaching methods. These differences make preparatory mathematics more suitable for students who want to further study mathematics in universities.