General > General Technical Chat
Calculus 1
Cujo:
--- Quote from: Cerebus on November 01, 2020, 10:23:28 am ---Presumably we're talking about an undergraduate (university) level course here?
--- End quote ---
It's a undergraduate Calculus 1 course.
I've seen over the internet and other university's that the topics that I have highlighted are from Calculus 2 and/or a Differential Equations course(s). So I think that my university crammed those topics into a Calculus 1 for whatever reason. |O
Cerebus:
--- Quote from: rstofer on November 02, 2020, 01:57:45 am ---Of course there will be a group of over-achievers who finish at least Calc I in high school. I think finishing the college equivalent of Calc II (Integral Calculus) in high school is a stretch. Maybe...
--- End quote ---
Yet that was regarded as the norm here in Britain when I did my maths A-level back at the end of the 70s. I don't know about the whole of 'calculus II*', but we were expected to have at least a good grasp of Integral calculus and be able to symbolically integrate expressions of real variables of reasonable complexity and know how to do numerical integration. As I've said, the dividing line between 'school' and 'university' seemed to be drawn just before you started doing differential equations.
To give you a flavour of what was expected, here's half a question from a maths A-level** paper from 1980 that's here
I've rendered half of one question here in mathjax to save folks from having to download and read that.
Here's the first of two parts of question 7. This is the easy part, part (b) is harder but I am not taking the time to render that into mathjax, it's way too long - see the pdf if you're curious.
7 (a) It is known that, for any integer \$k\$, \$\int^\pi_{-\pi} sin(k x) dx = 0, \text{and} \int^\pi_{-\pi} sin(k x) dx = \begin{cases}
2\pi & if k = 0 \\
0 & if k \neq 0
\end{cases}\$
Using the above results, show that if \$m, n\$ are positive integers,
(i) \$\int^\pi_{-\pi} sin(m x) cos(n x) dx = 0\$
(ii) \$\int^\pi_{-\pi} sin(m x) sin(n x) dx = \begin{cases}
\pi & if m = n \\
0 & if m \neq n
\end{cases}\$
(iii) \$\int^\pi_{-\pi} cos(m x) cos(n x) dx = \begin{cases}
\pi & if m = n \\
0 & if m \neq n
\end{cases}\$
* A bit of a foreign concept to a Brit, this uniform naming of undergraduate course units (and similar content between institutions) wasn't a 'thing' at British universities. At least, not in my day - I don't know about now.
** A-levels are the school examinations that Brits take at age ~18, covering material that was studied from ages ~17-18.
westfw:
In the US, it'd be sort-of one semester of Differentiation, one of Integration, and differential equations the next year.Except that differentiation and integration would be somewhat mixed. For example "AP Calculus" - a year-long class given in high school - would have "AB" versions that covered D&I, and a "BC" version that would be pretty much "more advanced D&I and maybe Taylor series."
It was a great time in college, when the calc classes, the physics classes, the chemistry classes, and the engineering classes, were all doing more or less the same math...
--- Quote ---Taylor series is specifically important to get in early, since it is used in a ton of theorem proofs later on.
--- End quote ---
Um... math majors would probably be expected to have different (more theoretical) classes than engineering/science majors.I don't recall every being particularly exposed to proofs at all (which is a weakness that hurts every time I try to take an "advanced" online CS class :-( )
Rick Law:
--- Quote from: Cujo on November 01, 2020, 07:09:59 am ---I have highlighted the topics in the Calculus 1 class at my school. I just want to know if those topics are normally taught in Calculus 1?
--- End quote ---
What is in Calc1 and Calc2 varies from school to school. It also varies depending on the student's major. I have some details below, but first some background on "two tracks".
Some universities have two separate 100 level (ie: first year) tracks for calculus: a more in-depth one for engineering and science majors (seeking BS degree) and the other for the less math-utilizing majors such as Economy/Business (seeking BA degree). The university I attended has two separate 100 level tracks for Physics and Chemistry as well - an in-depth tracks for engineering/science major. So multiple tracks is not uncommon.
In older days (less than a decade ago), even high schools have A and B class with A path going more in-depth.
Below are two links to "AP Calculus" with a lot of details. AP is "Advanced Placement" for high school students to earn some college credits while in high school. Since the high school student likely don't know what university they may be attending in the future, AP is therefore designed to be able to satisfy as many universities as feasible meaing the AP credits are accepted and counted as qualified credits towards graduation at the receiving university. So, (at least for the USA) AP is probably a good gauge to compare against to decide if your university has more, or not enough.
Note also that AP Calculus also have two tracks, AB and BC, one more in-depth than the other. You may want to review the links below and see if your university covers enough. If not, you can always supplement it by taking additional courses. Math and Physics in my opinion are great ways to practice and learn analytical skills.
Course at a Glance (4 pages) This is a good curriculum overview for you to compare against.
https://apcentral.collegeboard.org/pdf/ap-calculus-ab-bc-course-a-glance-0.pdf?course=ap-calculus-ab
Course details and examination details (248 pages deep dive):
https://apcentral.collegeboard.org/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf
george.b:
Here in Brazil, at least at my university, we have differential equations in Calculus II. All of the rest we see in Calculus I, Taylor series included.
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