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What should I expect from upper level math?

I love math and I definitely see myself involving it in my career. How does upper-level mathematics, specifically calculus, differ from math I have encountered in high school, my highest being AP Calculus? And which paths have more challenging math?

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Kris’s Answer

Its exciting to find someone so interested in mathematics. Beyond Calculus, probability and statistics, Linear Algebra (matrices) and differential equations comes more abstract theories. What do you want to do as a career? Or do you just love academia life? Physics and Astronomy can really use people who can fully grasp mathematical concepts. There is so much about our universe that we do not yet understand, and understanding mathematics is the key to unraveling those mysteries.

Probabilty and statistics is its own separate branch that is difficult for many to master. Consider being an actuary if this is your forte.

Many of us peak out, once mathematics becomes too abstract. I encourage you to continue to challenge yourself, ask questions, go as far as you can.

Kris recommends the following next steps:

read a biography about a famous mathematician
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Andrew’s Answer

AP Calculus AB can be articulated into the traditional STEM-track Calculus I, whereas AP Calculus BC into Calculus II for many colleges, provided you score at least 4 in the AP courses.

There are different levels of involvement in mathematics in various career paths. You will be a “user” of mathematical tools if you are in the field of science and engineering. If you are a mathematician, you are working on the further development of mathematics, either for the sake of its beauty or for the service of science and technology.

In any case, mathematics is a challenging field. It takes commitment and dedication.

As far as Calculus is concern, it should be appreciated that calculus is just part of the preparatory path to get ready for the study of mathematics. It is in Calculus that we study behavior of functions beyond the level of college algebra with Calculus I in local behavior, and Calculus II in global behavior. A subsequent course will be Differential Equations, in which Calculus I & II are just tackling the simplest cases of ordinary differential equations. After this course, you have completed the preparatory courses for the study of functions. The subsequent upper-level (400 level) course is Real Analysis, in which you will learn Calculus again for the first time. This is the first course you will have a taste of mathematics.

What is discussed above is only about functions. In mathematics, you will need to learn about the mathematical systems. It is where you start out with Linear Algebra and completed with an upper-level (400 level) course in Abstract Algebra, in which you study the foundation of algebra.
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David’s Answer

Calculus seems to be the key class for most everything. I believe there were about 3 semesters of calculus to get through. The more you can take or understand before you get to college is helpful. In some cases, even if you have taken certain AP math classes in high school they do not always go one for one with the same class in college. If there is time for example you may take the college calculus class even if you took one in high school and could skip it, especially if it the basis for so much more. As you move up the math classes they get more specific and detailed and to be honest I wondered why even had to take them. One class I recall was called Matric Analysis and I am pretty sure I have never used it since. To me the best part of math and understanding it was that it teaches you how to really think and solve problems.
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Natalia’s Answer

As you go up the Math ladder, especially on the postsecondary level, Math can get more and more abstract in comparison to High School Math. Another thing that changes in the introduction of proofs, which means you'll learn why a statement/formula/theorem is true and won't take it for granted anymore, you'll also learn how to prove up to a certain point, this happens even in Calculus.
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