Math 21 before Math 51?

Taking Math 21 before multivariable calculus

The content of Math 21 (improper integrals, infinite series, and power series) is essentially the material of BC-level AP calculus not in the syllabus of AB-level AP calculus, nor is it in IB Higher Level math. The math placement diagnostic results do not waive Math 21 requirements, since the diagnostic has no exam security; its feedback is purely advisory.

Credit for Math 21 is an enforced prerequisite to enroll in Math 51 or CME 100. Such credit can also be obtained via a score of 5 on the BC-level AP calculus exam, earning a suitable score on certain international exams, or transferring in credit for it from a suitable course at another post-secondary institution.

This is an enforced prerequisite for multiple reasons:

1. Math 21 material arises in throughout university-level quantitative modeling, such as in probability & statistics (so machine learning and data-intensive experimental scientific work), financial models, and differential equations (Math 53 and beyond, also useful in solving some optimization problems in AI as well as diffusion models).

   A former Math 21 student said:
   “…there is no hope of you being a minimally decent engineer/physicist if you don't know how to do any of these things."

    

   The same applies to doing minimally decent quantitative work in many fields.

2. Knowledge of this content keeps your options open among quantitative majors.
3. At all peer institutions, students are expected to learn this fundamental material before taking multivariable calculus.

4. A Stanford student who did well in math in high school by rote learning once wrote:

   “my foundational math skills were poor and I struggled [with] new material [in other fields] more than I should have. [...] I [spent] a summer break working through calculus books to make up for it. Don't make the same mistake I did: put in the time to learn the material properly. It will pay for itself many times over."

    

   The purpose of prerequisites is not merely to fulfill a requirement; it gives you the skills to apply what you learn in a confident and reliable manner.

5. Math 21 homework gives students practice writing more complex solutions than they ever see in Math 20's syllabus (AB-level AP calculus). Those higher-level skills improve your overall thinking about math, and thereby reduce the time commitment for more advanced math (such as multivariable calculus).
6. The demands of a 5-unit course (such as Math 51) are a significant ramp-up from a 3-unit course (such as Math 20), especially if the latter was at least a partial review for the student, since then the student had a cushion (in Math 20) that will not exist for them in the succeeding course. So it is a more gentle adjustment to go through Math 21 (a 4-unit course) before Math 51.