Can we use the tangental point to find gradients?

[lioric]

TL;DR

A student of mine came and said that their teacher said that they were to use the coordinates of the point of tendency as one of the coordinates to find gradient.

We were taught to take coordinates like this

But teacher is telling the student to take coordinates like this. What are the major reasons why this is not taught like this. I know the value would be the same, but I also know there is a reason why we don't use this method.

 

[PeroK]

What's the difference?

 

[lioric]

What's the difference?

I don't see one. We get the same answer. But we were taught that it was bad practice to use the point of tangency as one of the gradient coordinates. We were asked to take two points further from the point of tangency. And still all the tutorials and books show that the two points are taken excluding the point of tangency, like shown in the first diagram

 

[PeroK]

I don't see one. We get the same answer. But we were taught that it was bad practice to use the point of tangency as one of the gradient coordinates. We were asked to take two points further from the point of tangency. And still all the tutorials and books show that the two points are taken excluding the point of tangency, like shown in the first diagram

It's clearly a bad idea to draw the line only as far as the tangent point as this increases the margin for error. And, it's probably better to use all of the tangent line you've drawn. But, the more significant error, I assume, comes from drawing the line.

 

[lioric]

It's clearly a bad idea to draw the line only as far as the tangent point as this increases the margin for error. And, it's probably better to use all of the tangent line you've drawn. But, the more significant error, I assume, comes from drawing the line.

I suppose in an ideal exam situation, you can get away with it, but it's going to give a huge margin of error when you actually draw a graph from an experiment with best fit line.
Hence why it's bad practice.
I suppose teachers are just teaching to score in an exam now, not to learn

 

[lioric]

It's clearly a bad idea to draw the line only as far as the tangent point as this increases the margin for error. And, it's probably better to use all of the tangent line you've drawn. But, the more significant error, I assume, comes from drawing the line.

Here the coordinates are taken far from the tangent, but one is from the point of tangency.
Would this be ok? I still feel we should extend the tangent to the other way and take from both sides. And avoid using the point of tangency

 

[PeroK]

I don't know Desmos. Is that a freehand drawing?

 

[lioric]

I don't know Desmos. Is that a freehand drawing?

It's not a freehand drawing.
I'm just asking, if in a freehand graph, is it ok to take two points like this. One far away, the other, from the point of tangency ?

 

[PeroK]

It's not a freehand drawing.
I'm just asking, if in a freehand graph, is it ok to take two points like this. One far away, the other, from the point of tangency ?

I don't see why not.

 

[lioric]

I don't see why not.

Ok. Thank you. Much appreciated

 

[Steve4Physics]

TL;DR Summary: A student of mine came and said that their teacher said that they were to use the coordinates of the point of tendency as one of the coordinates to find gradient.

To minimise the uncertainty in the gradient, the triangle should be as large as convenient. This will usually mean the tangent’s point of contact is not at vertex of the triangle.

(If the reason for this isn’t clear, consider the uncertainty in the gradient if the triangle is very small.)

In addition, when constructing the triangle it helps (slightly) to place the triangle’s corner (the right-angle) at a convenient coordinate , e.g. on (5, 2) rather than (5.15, 1.95).

Well, that’s what I was taught (and is what I taught my students).

 

[lioric]

To minimise the uncertainty in the gradient, the triangle should be as large as convenient. This will usually mean the tangent’s point of contact is not at vertex of the triangle.

(If the reason for this isn’t clear, consider the uncertainty in the gradient if the triangle is very small.)

In addition, when constructing the triangle it helps (slightly) to place the triangle’s corner (the right-angle) at a convenient coordinate , e.g. on (5, 2) rather than (5.15, 1.95).

Well, that’s what I was taught (and is what I taught my students).

Thank you very much. I do it as well. And I teach it just the same. I was taught like that too. Just wanted to know the reason for it. My hunch was on uncertainty too. Thank you very much