On A Textbook's Explanation Of Derivatives And Limits

Question

For the past few days, I've been struggling to understand the concepts of tangent lines and instantaneous velocity. I picked up a textbook about the topic ("The Complete Idiot's Guide To Calculus"). This is what it has to say about how secant lines, that measure the gradient between 2 points of a curve, become tangent lines, that measure the slope of a single point on a curve:

I have 3 questions regarding this:

1. Does this mean that the tangent line isn't actually a line that is derived from the gradient of one point? It is instead the gradient over 2 points infinitely close together, so close that the distance between each other is delta x? Basically, so basically, the tangent line's gradient is given by the value of the limit? Am I right saying that?

2. For a distance time graph, where the derivative = instantaneous velocity, does that also mean that instantaneous isn't truly measured over a time period of 0, but instead a time period of $\Delta x$, where $\Delta x\to 0$? So instantaneous velocity is the value of the limit? But not actually the value given when distance is measured over a time period of 0?

3. Would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation? Because both of these are meant to be over a single point, over a time period of 0. But here, we are avoiding the nasty division by 0 by finding it over delta x instead, a value so small that is effectively 0, but not 0 (so we don't need to divide by 0)? So would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation?

Can all 3 questions be answered not too rigorously, without epsilon delta proofs?

Answer 1

To some extent these are a matter of interpretation or even opinion, but I'll give you my take.

1. In the development you are reading, the tangent line is determined by the gradient at one point, and the approximation through the average gradient between two points is made to come up with the definition. On the other hand, over the years, many people have thought of it as a an infinitesimal change in $y$ divided by an infinitesimal change in $x$, and you are free to do so if this is more intuitive for you. This approach was made rigorous back in the $1960\text{'s}$ by Abraham Robinson.

2. I would agree with this, at least in a practical sense. How could the velocity at single instant ever be measured?

3. The instantaneous speed is, I think an approximation, for the reason I gave in 2. On the other hand, the tangent line is a mathematical construct, with a rigorous definition, so I wouldn't call it an approximation.

Answer 2

1. Does this mean that the tangent line isn't actually a line that is derived from the gradient of one point? It is instead the gradient over 2 points infinitely close together, so close that the distance between each other is delta x? Basically, so basically, the tangent line's gradient is given by the value of the limit? Am I right saying that?

Note that the derivative at $c$ $$f'(c)=\lim_{\Delta x\to0}\frac{f(c+\Delta x)-f(c)}{\Delta x}$$ by definition. Hence it is the same.

2. For a distance time graph, where the derivative = instantaneous velocity, does that also mean that instantaneous isn't truly measured over a time period of 0, but instead a time period of delta x, where delta x -> 0? So instantaneous velocity is the value of the limit? But not actually the value given when distance is measured over a time period of 0?

Yes, you are right. It is the value of the limit. You cannot measure velocity over a time period of 0 since division by 0 is not possible.

Would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation? Because both of these are meant to be over a single point, over a time period of 0. But here, we are avoiding the nasty division by 0 by finding it over delta x instead, a value so small that is effectively 0, but not 0 (so we don't need to divide by 0)? So would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation?

Yes, the slope of the tangent line respectively the “instantaneous” velocity can be approximated by the difference quotient giving the slope of the secant respectively the average velocity.

Answer 3

1. Does this mean that the tangent line isn't actually a line that is derived from the gradient of one point? It is instead the gradient over 2 points infinitely close together, so close that the distance between each other is delta x? Basically, so basically, the tangent line's gradient is given by the value of the limit? Am I right saying that?

Yes. We start with a heuristic understanding of what the tangent line is, but no definition. We then draw lines which are “close” to the tangent line in that they share one point with the tangent line, but are secant lines instead. We can measure the slopes (gradients) of those secant lines. If the slopes have a limit as the points approach each other, we define the tangent line to be the line with that slope. So the slope comes first, then the tangent line.

2. For a distance time graph, where the derivative = instantaneous velocity, does that also mean that instantaneous isn't truly measured over a time period of 0, but instead a time period of delta x, where delta x -> 0? So instantaneous velocity is the value of the limit? But not actually the value given when distance is measured over a time period of 0?

The point is that there isn't any “value given when distance is measured over a time period of 0.” An object travels zero distance in zero time. Instead, instantaneous velocity is a limit of average velocity over smaller and smaller intervals of time.

3. Would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation? Because both of these are meant to be over a single point, over a time period of 0. But here, we are avoiding the nasty division by 0 by finding it over delta x instead, a value so small that is effectively 0, but not 0 (so we don't need to divide by 0)? So would it be a stretch to call the tangent line and the instantaneous speed as given by the difference quotient an approximation?

The (slope of the) tangent line and the instantaneous speed are limits, not approximations. They are approximated by the slopes of secant lines, or average velocities, over small intervals. But they themselves are not approximations.