# Tag Info

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### How does one tutor an A-level student past the derivative paradox?

There is no royal road to geometry. - Euclid Nor calculus. The essence of calculus thinking is really the limit concept. One needs to wrap one's mind around that. Formally: it's the core technique ...
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### How does one tutor an A-level student past the derivative paradox?

I teach calculus at a community college in the U.S. (2 year college, from which many students transfer to a university). I explain limits from about day two in an informal way ("h gets infinitely ...
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### How does one tutor an A-level student past the derivative paradox?

I admit that I'm unable to follow the proof you give as an example in your question, but am I correct when I assume that your question simply wonders how to reconcile $dy/dx$ with the fact that $dx$ ...

### Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Here are some example questions. The graph of the function $f$ is given above. Evaluate the following limits. If the limit is infinite, write $\infty$ or $-\infty$ as appropriate. If the limit does ...
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### Do we need to practice equation derivation while learning math if equations will be chunked and automatized?

I endorse Steve's comment about different skills and different levels of proficiency in different skills. Also, if one at one time justified the Pythagorean theorem, but is no longer able to do so, I ...
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### A different symbol for the indefinite integral/antiderivative?

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 ...
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### When analytic form of derivatives is preferred over numerical form?

I think maybe if you try to write down precisely what you mean by $h<<1$ you will end up writing the definition for $\lim_{h \to 0}$. From this point of view, there is really no difference ...
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### Introducing derivative concept and definition

It occurs to me that maybe you can approach this from the viewpoint of approximations. Briefly, I’m thinking of the kinds of approximations they likely use in practice, and which are often given in an ...
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### How does one tutor an A-level student past the derivative paradox?

First off - I 100% agree with Collins here that there's no shortcuts. To really understand how derivatives work, you need to learn the epsilon-delta definition. But it's my sense that you're not ...
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### Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'll argue that this will be likely not feasible for a test question. There's several points in the calculus progression where there's a "hard bottleneck" of some sort, but once you get past ...
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### Interpreting the derivative as instantaneous rate of change in real phenomena

Shades of Zeno: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes And I don't think your confusion really has to do with calculus per se, but with the definition of speed (or velocity if we are using ...

### How to explain what's wrong with this application of the chain rule?

This is a VERY VERY typical problem. In fact, it's a problem even for $\frac{d}{dx}3^x$, much less your example. The way I try to deal with this is one of two ways. What has to happen first? To ...
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### Is there a point at which it makes decidedly more sense to learn about a "linear approximation" to a function, rather than a "tangent"?

Thinking about tangent lines as linear approximations is a great way to foreshadow Taylor series. After linear approximations in calc 1, I usually give students a few prompts to get them thinking ...
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### Are there direct practical applications of differentiating natural logarithms?

I couldn't find a lot either. Suggest playing with some logarithmic properties and constructing problems based on that. E.g. pH is log10 of the hydronium ion concentration. Could ask how the pH ...
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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 ...
I normally go other way around. Start with a logarithm, which by definition maps multiplication to addition: $\ln{ab} = \ln{a} + \ln{b}$. A detour into a history could be also useful: multiplication ...