New answers tagged calculus
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I'm not in education, so take my answer with a grain of salt. I am merely approaching this from a perspective of someone who is really frustrated with how mathematics is taught.
Mathematics is a tool we use to understand the universe. If you want to make math more engaging, it would be useful to actually show how one could use math to solve problems.
For ...
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It's about the learning experience.
An issue with project based learning is the need for objective grading isn't going away. Since the first semester of mathematics is what serves to filter out under performing students there is a desire to have an unmovable bar. Without it you get grade inflation.
This isn't to say there can't be projects. You just have ...
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I'm an engineer who really likes and uses mathematics on work and daily life.
Myself, I would try to make the course as much self-contained in classes, but still stick to some written material the students have access to. This allows the subject to be more complex and extensive while keeping down the difficulty settings.
I'd prove theorems as much as ...
1
Select and use an open textbook. Students who wish to use a physical textbook will be able to get a copy run off and hole-punched at a local print shop, then put in a binder, for far less than the cost of a conventional textbook.
You will save your students a pile of money and there are enough available options that you should be able to find something ...
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Without directly answering your question, you don't seem to have the background you need to be "improving" the undergraduate experience yet, and have some work to do. I think you're right in sensing that your question is too general
Don't talk about the "difficulties in forming such a course", and spend your initial time finding out what aspects of the ...
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Given 100% control, I would have one-to-one instruction. One instructor meeting individually with each student. That instructor can change the approach, the speed, the order of topics, the method of instruction, based on that individual student.
But of course hiring (and training) enough instructors for that is probably way beyond any reasonable budget. (...
3
You could show that, if $A^{ix} = f(x) + i g(y)$ is defined in a way that it satisfies $A^{ix+iy} = A^{ix} A^{iy}$ and other standard properties of the exponential function, then $f$ and $g$ must satisfy the addition and subtraction formulas for cosine and sine, so they have to be $\cos(kx)$ and $\sin(kx)$ for some $k$.
But showing that for $A = e$, the ...
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Can you define $e^x$ for real $x$ without calculus? You need to take a limit, I think.
If you allow "informal limit reasoning", then there is such an argument.
One way to define $e^x$ for real $x$ is by the limit $(1+\frac{x}{n})^n$ as $n$ grows without bound. This is often motivated with a compound interest story.
Substituting $i\theta$ in for $x$ into ...
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I believe Euler's identity can be reached via De Moivre's Formula:
$$\cos(nx)+i\sin(nx)=\left( \cos(x)+i\sin(x)\right)^n$$
(I am not finding a clear exposition of this route, as often Euler's identity is used to prove De Moivre's, whereas here we're seeking the reverse.)
Wikipedia says, "The truth of de Moivre's theorem can be established by using ...
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There is a proof of de Moivre's formula
$$
(\cos\theta + i \sin\theta) ^n = \cos(n\theta) + i \sin(n\theta), \qquad n \in \mathbb Z
$$
by induction (for $n > 0$) and symmetry. Maybe that is the best we can do without calculus.
Some textbooks (not assuming calculus) use a notation $\mathrm{cis}\;\theta$ meaning $\cos\theta + i\sin\theta$ and do all ...
2
I mainly teach physics, only a little math on the side. If I was teaching this math topic and wanted to spend 3 minutes giving the appropriate physical motivation, I would do something like the following.
There is something called energy. It comes in various forms. Food has energy, which is what we're talking about when we talk about calories. Heat, sound, ...
0
I would start with the scalar. Work = force times distance. Starting with the scalar makes things easier, more intuitive. (Not a math logic point, just a psychological one.) You can then easily generalize, perhaps without showing a derivation and say "well this is what work is in these trickier situations (vectors, non constant force, more dimensions, ...
2
The best way to illustrate the sense of $W=|F||AB|\cos(\alpha)$ is to show that it makes sense that each of the factors is proportional to the work done.
If you double the amount of force to an object to move it a certain distance, it doubles the amount of work needed.
If you double the distance to move an object by applying a certain force, it doubles the ...
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We observe the proof on Wikipedia. The intuition behind it is Cauchy's mean value theorem, which states that
$$\frac{f(a)-f(b)}{g(a)-g(b)}=\frac{f'(c)}{g'(c)}$$
for some $c$ between $a$ and $b$. From here all we need to do is divide by $g(a)/g(a)$ to get
$$\frac{\frac{f(a)}{g(a)}-\frac{f(b)}{g(a)}}{1-\frac{g(b)}{g(a)}}=\frac{f'(c)}{g'(c)}$$
and let $a\to ...
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If you teach the chain rule with Leibniz notation I recommend this suggestion of Steven Gubkin. It makes computations more explicit and straightforward, and students pick it up fairly well in my experience.
For the remainder I'll address some of the subtleties involved with derivative notation, the function concept and how that relates to the chain rule.
...
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My opinion on this topic has vacillated a little through the years, but for the most part, I've always spent time on background information about mathematical and statistical information because I firmly believe that doing so encourages an increase in student understanding and recall.
I believe it also helps students feel less stressed in my classes, ...
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The intuition here is basically that of Ben Crowell's answer, and that kind of intuitive explanation might be worth going through first. What I want to show is the kind of activity you can explore with students to investigate how it works in a "less than completely obvious" situation, when at least one of the rates of change itself keeps changing.
One way ...
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What is difficult about the chain rule is the function concept, more specifically the composition of functions. Notation that hides or leaves implicit the composition of functions causes a great deal of confusion for students. However, the fundamental issues are not the notation used (all choices are messy to some extent), but what the use of the notation ...
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I just start with constant rates of change, where it's pretty blazingly obvious that the chain rule works. E.g., Jane hikes 3 kilometers in an hour, and hiking burns 70 calories
per kilometer. At what rate does she burn calories?
Our class is generally much more comfortable with the $f'(x)$ notation, and as a result I stayed away from the $\frac{dy}{dx}=\...
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