31 votes
Accepted

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 ...
Daniel R. Collins's user avatar
31 votes

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 ...
Sue VanHattum's user avatar
  • 20.3k
30 votes

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

The root of the difficulty is that $x$ appears free in $f(z)$, but we are trying to "capture" it with $g(x)$, which is illegal. When we substitute $g(x)$ into $f(g(x))$, we have a variable clash: $$ f(...
Kevin's user avatar
  • 586
26 votes
Accepted

Finding the Balance in a Math Question (Teaching)

I think it is helpful to let students know that you are looking for their thinking while problem-solving, and not just answers. Then you can ask questions like: Find all of the points on a circle of ...
Carser's user avatar
  • 798
26 votes

A visualization for the quotient rule

Depending on how much algebra you allow, you could make the exact same rectangle picture but label the sides $g(x)$ and $q(x)$ with area $f(x)$. This geometrically enforces $g(x)q(x) = f(x)$, aka $q(...
Steven Gubkin's user avatar
19 votes

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

We routinely get questions related to pushing more rigor in early calculus. Usually from outstanding students and based on sample of one I like it that way logic. There's a reason why things are the ...
guest's user avatar
  • 207
19 votes
Accepted

What are some examples of great functions that are not too elementary (easy)?

I'm personally a fan of simple examples: $x e^x$ (has nice critical point, point of inflection) $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics) $\frac{x}{x^2+1}$ (a ...
TomKern's user avatar
  • 4,347
18 votes

Are there direct practical applications of differentiating natural logarithms?

Have you thought about the fact that you’re asking this in the middle of a pandemic for which log plots are being used all over the place to visualize the growth of COVID cases? At any rate, $${d \...
user1815's user avatar
  • 5,545
14 votes

Are there direct practical applications of differentiating natural logarithms?

Whenever we measure a quantity on a log scale (such as Richter, decibels, musical pitch, or a log-plot axis), we are focusing attention on relative variation in that quantity. If $y = \ln x$, we have $...
nanoman's user avatar
  • 271
13 votes

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

f is not a function of (only) z - f here is a function of x as well as z. I think this explanation is intelligible to a calc 1 student, and gets at the heart of the matter.
Henry Towsner's user avatar
12 votes

A different symbol for the indefinite integral/antiderivative?

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 ...
JTP - Apologise to Monica's user avatar
12 votes

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

Start with a numerical example. Say you want to find the gradient of the tangent to $y=x^2$ at $x=1$. Obviously the point itself is $(1,1)$. Pick a nearby point, say $x=1.1$. A moment with a ...
Nullius in Verba's user avatar
12 votes

What are some examples of great functions that are not too elementary (easy)?

Don't forget purely graphical problems. Give a graph of $f$ or $f'$, ask for the student to sketch the other one.
Steven Gubkin's user avatar
9 votes

Finding the Balance in a Math Question (Teaching)

Note: It's possible that in the future, Wolfram Alpha will improve and be able to answer the questions in this answer, so it's best to actually try them in Wolfram Alpha first. Use questions that ...
JRN's user avatar
  • 10.8k
8 votes
Accepted

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

$$ \frac{d (3^{5x+1})}{dx} = f'(g(x))g'(x)= \frac{d \left(3^{5x+1}\right)}{d(3)} \times \frac{d (3)}{dx}. $$ However $\dfrac{d (3^{5x+1})}{d(3)}$ is undefined.
Taemyr's user avatar
  • 228
8 votes

Why most people think that :$(fg)'=f' \cdot g'$?

As mentioned, a probable cause is an implicit reasoning as if every operation were a homomorphism (similar to the implicit reasoning by linearity). Similar errors include $\ln(x+y) = \ln(x)+\ln(y)$, $...
Benoît Kloeckner's user avatar
8 votes

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 $dx$ ...
Peter - Reinstate Monica's user avatar
8 votes

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 ...
Steven Gubkin's user avatar
8 votes

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 ...
Will Orrick's user avatar
  • 1,122
7 votes

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 ...
Aeryk's user avatar
  • 8,011
7 votes
Accepted

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 ...
Chris Cunningham's user avatar
7 votes

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 ...
Dave L Renfro's user avatar
7 votes

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 ...
Reese Johnston's user avatar
7 votes

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 ...
Daniel R. Collins's user avatar
7 votes
Accepted

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 ...
guest philosopher's user avatar
6 votes

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 ...
kcrisman's user avatar
  • 5,976
6 votes
Accepted

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 ...
Jordan's user avatar
  • 603
6 votes

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 ...
guest's user avatar
  • 159
6 votes

A different symbol for the indefinite integral/antiderivative?

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 ...
Chris Cunningham's user avatar
6 votes

Exponential & logarithm in a high school calculus class

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 ...
user58697's user avatar
  • 171

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