# Tag Info

• 4,619

### 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 ...
• 2,089

### 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 ...
• 177

• 8,751

### 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 ...
• 20.5k

### 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 ...
• 743

### 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 ...
• 5,184
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.
• 210
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 ...
• 19.1k

### 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 ...
• 1,774

### 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 ...
• 21.1k
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 ...
• 593

### 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 ...
• 149

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

Give a function together with some properties of the function, but do not give a formula, and ask for the derivative. An example: Let $f: \mathbb{R} \to \mathbb{R}$ be a function which enjoys the ...
• 20.5k

### Are there direct practical applications of differentiating natural logarithms?

A couple of direct applications: Showing that a power law appears on a log-log graph as a straight line with gradient equal to the exponent of the power law (although that can be done by other, ...

### Are there direct practical applications of differentiating natural logarithms?

Boltzmann's equation for entropy is $S=k\ln W$, and the second law of thermodynamics is all about change in entropy. Maybe this is a place to start with your quest for a practical application of the ...
• 7,703

### In single variable calculus, do you distinguish between critical and singular points?

The main reason we have the word "critical point" is for the first derivative test. The statement is slightly easier if you only have to say "critical point" instead of "...
• 20.5k
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 ...