# Introducing derivative concept and definition

I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate place in first quarter calculus to support the delivery. In particular, they would have been introduced to the limit, be able to compute limits for many interesting examples, had a treatment of continuity and understand what a removable discontinuity is.

So the general question here is, just take a time-tested standard treatment from a textbook, or try to do something fancier or outside the box? More particularly, introduce it as a textbook would (derive the definition and then compute examples), or jump right into computation using the definition of derivative, and then "motivate" or derive this definition?

• Is there a reason for simply not looking through their engineering textbook(s) and picking things specifically dealt with there? Picking a book I got back in 1980 --- Irving H. Shames' Engieering Mechanics, 3rd edition --- the first few pages of "Volume II. Dynamics" reviews differentiation and gives some examples involving projectile motion. I would simply go to the university library and look through beginning engineering and "mathematics for engineers" books, then prepare the presentation. May 5 '19 at 10:29
• By the way, "derive the definition" seems to be incorrectly worded. I would stick with "motivate the definition". And before I leave (I need to go somewhere in a few minutes), this list of Stack Exchange Mathematics Educators questions, as well as this other list, may be worth looking through. May 5 '19 at 10:43
• Can you offer a bit on why the regular intro to the derivative isn’t working for You? May 5 '19 at 20:25
• Given that "In particular, they would have been introduced to the limit, be able to compute limits for many interesting examples, had a treatment of continuity and understand what a removable discontinuity is," seems to serve as a good "bridge" to the definition of the derivative. In particular, help bridge their current understanding and experience with solving limits, by giving a couple of examples of limits in which each, in fact defines the derivative of a particular function, and share that what you've all just done was to find the derivative. Etc. May 5 '19 at 20:48
• The obvious motivation is a good one, which is seeking to find a way to calculate instantaneous rates of change, e.g. the speed on a car speedometer. Transitioning from calculating average rates of change via slopes of secant lines to calculating instantaneous rates of change via the slope of a tangent line, and then motivating the definition of a derivative as a limit that gets you an instantaneous rate of change by determining a slope of a tangent line via a limit, is tried and true motivation (e.g., Stewart calculus). May 9 '19 at 2:53

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 appendix at the back of texts, such as the following:

For $$x \approx 0$$ we have $$\frac{1}{1-x} \approx 1+x$$ (multiply numerator and denominator of left side by $$1+x$$ and then drop the $$x^2$$ term).

For $$\theta \approx 0$$ we have $$\sin \theta \approx \theta$$ and $$\tan \theta \approx \theta$$ (use a geometric argument).

For $$\delta \approx 0$$ we have $$\sqrt{a^2 + \delta} \; \approx \; a + \frac{\delta}{2a}$$ (see here for an algebraic derivation).

Both the product rule for derivatives (expand $$(u + \Delta u)(v + \Delta v)$$ and drop the $$“\text{small} \times \text{small}”$$ term) and the quotient rule for derivatives (multiply numerator and denominator of $$\frac{u + \Delta u}{v + \Delta v}$$ by $$v - \Delta v$$ and drop the two $$“\text{small} \times \text{small}”$$ terms) can be heuristically obtained by methods that they will be (or at least should be) familiar with. The idea is that you want the simplest non-constant approximation, which winds up being a linear (first degree) approximation, and this arises as the linear part when you are able to expand the quantity as a (or some simple algebraic combination of) possibly infinite sum(s) of non-negative integer powers of “small things”. Then maybe briefly discuss aspects of (a) why we can drop the higher order small terms (this graphical exploration idea is worth considering) and (b) the value of a linear approximation (easy to integrate linear functions, can apply proportionality when linear, simple to numerically calculate or estimate when linear, etc. --- note that I’m trying to avoid simply saying “linear” is good or “graph locally looks like a line” is good, because this leaves unanswered why linear/lines are good/useful in the first place).

Finally, the books I list in my answer to Teaching Calculus Less Formally (especially Calculus Made Easy by Silvanus P. Thompson and Calculus in Context. The Five College Calculus Project by James Callahan, et al.) might be worth looking through for ideas in general.

• This is the example I would have given to this specific context. There are some specific examples of linear approximation in many introductory calculus books that can be directly related to the notion of "first-order approximation", such as from the ideal gas law vs. other gas laws or tension formulas. May 6 '19 at 17:02

I think you should take whatever motivation you are most excited to show them based on (a) their interests and background (b) your passion and go that way. Everyone will have a different answer to this and so it's important to, with careful consideration, do whatever you, the educator, feels is best.

For example, as someone with a geometric bent, I would ask how we could define that tangent line we all know how to draw, using only our techniques for finding the line between two points. Motivating the definition via pictures usually feels best given my inclinations and my students' preparations. But you know your students better than us.

A good approach for engineers might be to connect the "slope of a graph" to sensitivity: the factor by which a small change in input gets multiplied to produce a change in the output. You can connect this to both understanding how changes propagate through a system and error analysis.

You can either do this purely symbolically to derive the definition of the derivative (with h as your error), or start by analyzing the problem of sensitivity for linear functions, and then show that tangent lines are good approximations for a graph up close, and then derive the definition through secant lines approaching tangent lines.

I would go with a graphical interpretation to start with. I remember learning this sort of thing in the USN sub force for power transients on reactors (rods, reactivity, power). It was especially useful for the enlisted men since they had less math. But even for officers, I felt that it gave huge insight to have to draw curves (even by eye, phenomenological, directional) of derivative or integral. I had a 5 on AP BC (and a rocking 5, didn't encounter a single problem I couldn't solve). But still, I felt the visual interpretation was new, cool, useful. You actually draw the curves "above each other" (so you can connect the inferences). Definitely made me think more about the actual physics, than just some algevra-laced calculus.

And then after that, show a bit of math.

P.s. In addition to actually helping the kids, I think if you show something like this, visually, it will give the hiring committee that feeling that you actually care about the kids. And are a cool fish, pedagogically. Two homicided birds. ;-)

P.s.s. Don't do rod height...too geeky submarine. But maybe, pushing down the accelerator pedal on a car.

P.s.s.s. The hiring committee figures you have the props on algebra-based calculus. But show them a little creatitivity and outreach. Cha-ching, bay-bee@ Buy me a beer, if you get the gig.

• I think this answer would be much improved by having the meta-commentary removed. Mar 23 at 23:06