9th grader attempting to learn Calculus 1, 2, and 3 during the summer

Question

I am about to begin my extensive journey into higher and more advanced mathematics, and my starting point is learning calculus 1, 2, and 3. I prefer self-studying as opposed to a taking a class in school, (e.g. AP Calculus AB/BC) and as such, I would like to ask how you have learned Calculus 1, 2, and 3 in the past, what key concepts are necessary to understand, and what are the more difficult ideas/topics to understand in the subject. Thank you for your time.

Answer 1

I learned calculus while in high school by reading books from the public library. It is best to sample a variety of books. You can also get a lot from wikipedia.

You must acquire an understanding of the foundations of $\Bbb R$. It is possible to extend $\Bbb R$ to a larger ordered field which has "numbers" that are positive but less than any positive rational, but you cannot do standard calculus with such a field. Strangely, there was no thorough written definition of $\Bbb R$ before the 19th century, long after the development of calculus.

We show that we can extend the ordered field $\Bbb Q$ to an ordered field $\Bbb R$ which has the property that any non-empty subset with an upper (lower) bound has a least upper bound (greatest lower bound). Any such extensions of $\Bbb Q$ are isomorphic. This is why we speak of "the" reals.

$\Bbb Q$ does not have the lub/glb property. E.g. $\{ q\in\Bbb Q : q>0\land q^2>2\}$ has no $rational$ glb.

It follows that

$(1)$. If $0<r\in\Bbb R$ then there exist $q_1,q_2\in\Bbb Q$ with $0<q_1<r<q_2$.

$(2)$. If $r_1,r_2\in\Bbb R$ with $0<r_1<r_2$ then there exists $n\in\Bbb Z^+$ with $nr_1>r_2$. This is called the Archimedean property.

$(3)$. If $r_1,r_2\in\Bbb R$ with $r_1<r_2$ then there exists $q\in\Bbb Q$ and $r_3\in \Bbb R\setminus\Bbb Q$ with $r_1<q<r_2$ and $r_1<r_3<r_2$.

A basic understanding of limits (in general) and limits of sequences is necessary. (An infinite sequence $(a_n)_n$ is any function $f$ whose domain is $\Bbb N$ or $\Bbb N_0$ but we write $a_n$ for $f(n)$. But sometimes it is convenient to let the domain be some other infinite subset of $\Bbb N_0$). Different authors employ different notations for a sequence.

$(4).$ Definition. $a=\lim_{n\to\infty}a_n \iff \forall r>0 \,\exists n_r\in\Bbb N\,\forall n>n_r\,(|a-a_n|<r).$

Some basic tools of limits are

(5). If $a=\lim_{n\to\infty}a_n$ and $b=\lim_{n\to\infty}b_n$ and $r\in\Bbb R$ then $a+rb=\lim_{n\to\infty}(a_n+rb_n)$ and $ab=\lim_{n\to\infty}(a_nb_n).$ And if $b\ne 0$ then there are only finitely many $n$ for which $b_n=0,$ and if we exclude those $n$ then we have $a/b=\lim_{n\to\infty}(a_n/b_n).$

$(6).$ A useful tool that students often overlook is that $a=\lim_{n\to\infty}a_n$ iff the set $\{n\in\Bbb N : |a-a_n|>r\}$ is finite whenever $r>0.$

Caution: It is often convenient to augment $\Bbb R$ to $\Bbb R\cup\{\pm\infty\}$ but the usual rules of aritmetic do not apply to $\pm\infty$. For example, you may see $\infty=\lim_{n\to\infty}a_n$, which means $\forall r\in\Bbb R \,\exists n_r\in\Bbb N\,\forall n>n_r\,(a_n>r).$