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You are interchangeable, like peas in a pod. Cauchy and Weierstrass were usually saying "$f(x)$ becomes arbitrarily close to $L$", with the qualifier "as $x$ approaches $a$", sometimes before, sometimes after, sometimes implied. They also followed Leibnitz and Lagrange to talked about a quantity $f(x)$ becoming infinely close to $L$, when $|x-a|$ is ...


-1

It's the whole subject of the Bourbaki's book IV Integration. First generalize your notion of derivative: a derivative is any function $d:A \rightarrow A$ of the space $A=\mathcal{C} ^ \infty(\mathbb{R})$ such that $d(f+g) = d(f)+d(g) $ $d(\lambda f) = \lambda f $ ($\lambda \in C^{te}) \cong R \subset A$ $d(fg) = f d(g) + d(f) g $ The first two mean the ...


3

I'm teaching AP Calculus for the first time this year, and my students are all asking the same question as you. Here's what I tell them. If you're going to college next year, and if you will be required to take at least a semester of calculus there (which is a pretty safe bet for nearly everyone but do your own research), AP Calc will do one of two things ...


1

An alternate notation that I made up for derivative and antiderivative of $f$ are $f^{\downarrow}$ and $f^{\uparrow}$. I experimented with them when trying to explain or write steps of tableau method or ladder method for multiple integration by parts. (This is hallowed ground so I will be offending many, but I go ahead and drop the variable and its ...


3

Before inventing new notation, it is very important to learn the accepted notation and teach it to your students correctly. The question contains the following claim: $$\int \cos = \sin$$ which is not a meaningful statement. It is either incorrect, or unclear. This notational issue is not a nitpick: it is a critical issue that, if it goes uncorrected, ...


3

I'll echo other responses with the same: Do NOT introduce made up notation. I've made the effort in my Calculus courses to follow your outline while avoiding any new symbols. Similar to user20311's comment, when I first cover antiderivatives, all questions are phrased as either Find an antiderivative of $\sin(x)$, Find $F(x)$ such that $\frac{d}{dx}F(x) = ...


0

The closest approach (pun intentional) to a limit in precalculus is when you're dealing with asymptotes: https://youtu.be/gOQffI21rpc?list=PLKXdxQAT3tCuJku9nTlRZgx_RjGZ7djMc The advantage here is that it really drives home the idea of the limit of a function as a prediction of what the function value will be. Limits at finite values are harder. You ...


4

First, yes, many teachers use that term, "anti-derivative" and "integral" only when a definite integral is in use. For your examples of each, it seems to me that you offered a clear distinction between the two expressions, i.e. that the bounds is what what makes the integral definite. For me, that's where the explanation to students tends to reach a ...


3

"Solving (routine) problems fast" is a useful skill, but very far from what you really need. You have to be proficient in reading, understanding and criticizing proofs, come up with your own proofs. Be able to plan (and execute the plan) to attack ill-defined problems. Use tools, like a computer algebra system, Google or math.SE to ease over routine stuff/...


2

I then progressed to the point that I could manage to do 85% of all the questions in a minute. Rarely did I take 1.5-2 minutes (on the really difficult ones). It sounds like you were doing very simple, plug-and-chug problems. For example, evaluate the integral from x = -3 to x = 4 of (4x³+3x²)dx. Problems that require detailed proofs, or examination of ...


12

According to UCAR-10101: Handbook for Spoken Mathematics, Lawrence A. Chang, Ph.D., page 38, a source for how to speak mathematics to sight-impaired students, we have: $$\lim_{x\to a} y = b$$ is spoken as the "limit as $x$ approaches $a$ of $y$ equals $b$". For the given expression, $$\lim_{x\to a} f(x) = L$$ is spoken thusly: the "limit as $x$ approaches $...


3

As $x$ approaches $a$, $f(x)$ approaches $L$. First, we emphasize what is happening to the independent variable, then we explain the consequence. I think that this phrasing is concise and easy to understand. It is clean and efficient. This is essentially (3), but I think that the sub-clause "The limit..." is unnecessary. Moreover, if we have a fixed ...


7

I say it the third way, for these reasons: Firstly, from a notation point of view, the “$x\to a$” has to be written with the “$\lim$”, and no “$\lim$” can be written without it (without specifically saying what you mean by not having it), so it makes sense to put them together when you say it aloud. Indeed, you could argue that $\displaystyle\lim_{x \to a}...


3

I usually say f(x) lähestyy L:ää, kun x lähestyy a:ta and sometimes I instead say it more colloquially as f(x):n raja on L, kun x on a. The inverted Kun x lähestyy a:ta, f(x) lähenee L:ää is also fine and in use. These would correspond to 2) and 4) in English. I would avoid linguistic complexity, such as a side clause embedded in the sentence, ...


1

Precalc courses differ a lot, so it's hard to say something is included or not. After all there was a time when there didn't even exist a precalc course (different from a strong algebra 2, trig, and nalytic geometry sequence). Then again there are some precalc course that spend a reasonable amount of time on theory of functions (and relations), ...


0

1 > 3 > 2 I prefer the first wording. You get more of the idea of thinking about what x causes what y. Yes, the second is equivalent, but it is awkward, to add the condition (approaching a) after the result. Three is OK also. Although I mildly prefer 1. Maybe shorter.


5

If you want to find academic studies, do an academic literature search. Didn't you learn to do this during your studies? Even if the results show little, if you give us the parameters that you searched with, it helps others to respond/add. I also recommend you not to expect too much from formal education research. Teaching is a complex affair, full of ...


1

I'd say the concept of zero, infinity, and the infinitesimal. Often they're thought of like numbers and this leads to calculations which don't make sense where, even if they work numerically, it's not actually correct since the process is wrong. With calculus, I think this would be the idea of the limit and everything that revolves around it, which is ...


0

Perhaps what would best prepare a student for the formal $\epsilon{-}\delta$ definition of a limit is an animation which shows both $\epsilon$ and $\delta$ approaching $0$, $\epsilon \neq \delta$, something like this (apologies for the too-high speed):                     (Image from here.) (This would ...


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