Linearization and Differentials

[PFuser1232]

I was reading a chapter on differentials in my calculus book, when I came across the graph shown in the image attached to this post. Two questions came to my mind upon seeing this graph:
1) Isn't it technically wrong to label the x-coordinates as x and (x + Δx)? I mean, wouldn't it be more appropriate to label them as a and (a + Δx)?
2) I have always been under the impression that differentials are infinitesimally small. How then can a geometric definition in which differentials are treated as normal real numbers arise?

 

[mathman]

1. There is nothing wrong with calling it x and x + Δx. Why do you object?
2. The picture is misleading. Δx and Δy are differences. Labelling them dx and dy is incorrect.

 

[PFuser1232]

1. There is nothing wrong with calling it x and x + Δx. Why do you object?
2. The picture is misleading. Δx and Δy are differences. Labelling them dx and dy is incorrect.

My objection is based on the following:
Given a real number a, we can express the x-coordinate mathematically as x = a. How can we define an x-coordinate as x = (x + Δx)? If we define it in this manner, then we get Δx = 0 which makes no sense mathematically.

 

[bigfooted]

My objection is based on the following:
Given a real number a, we can express the x-coordinate mathematically as x = a. How can we define an x-coordinate as x = (x + Δx)? If we define it in this manner, then we get Δx = 0 which makes no sense mathematically.

When Δx = 0, then x=x, nothing wrong with that. But when working with infinitesimally small Δx, it will never actually reach zero.
Also, any point on the x-axis can be called "x", that's why it's called the x-axis. If you mean a specific point on the x-axis you could say that x=2 or x=a.

 

[TitoSmooth]

I was working on these yesterday. At a point x= some value (call it a) there is a tangent at that point. Remember what delta x means. The displacement of n object along the axis. Geometrically we can argue and say that x to x + delta is the length of the base of the triangle.[run]

To make it more clear. Let's say x=3 AND delta x=4. And x=3 is a point along the x axis.

So x will lie 3 points from the origiin to the right.

Now delta x plus x gives us 3+4= 7

So x+delta x=7 which is our new x coordinate so connect the two points.Do you want me to list the actual def? It can be explained with the difference quotient when the author starts talking derivatives but before you see differentiation.

 

[TitoSmooth]

Do you understand what linear it at ion is about tho? What's the difference between dy n delta y etc and why you are doing this?

I know you are using stewart and he batteries this section.

 

[verty]

There are many ways to interpret dx and dy, what they are. One can say they are infinitesimally small, but that really means "large enough not to equal 0 but small enough for the square to equal 0". So they are numbers small enough so that squaring them makes them zero. This is not rigorous but it is one way to think about it. But I personally don't recommend thinking in this way.

Another way is to think of dx and dy as rates of change, that is, time derivatives: $dx = {dx \over dt} = x'$, etc. This works pretty well and is how I recommend thinking about them.

Another way is to think of them is as component distances along the tangent: $dy = {dy \over dx} dx$. Some people prefer this and it is said to be a useful way of thinking for more advanced math.

 

[PFuser1232]

When Δx = 0, then x=x, nothing wrong with that. But when working with infinitesimally small Δx, it will never actually reach zero.
Also, any point on the x-axis can be called "x", that's why it's called the x-axis. If you mean a specific point on the x-axis you could say that x=2 or x=a.

Exactly. Any point on the x-axis can be called x. But not, say, x+3. Because when we do that we'll end up with something like:
x = x + 3 ⇔ 0 = 3

 

[mathman]

Exactly. Any point on the x-axis can be called x. But not, say, x+3. Because when we do that we'll end up with something like:
x = x + 3 ⇔ 0 = 3

I feel you are making too much out of a notation question. When discussing x and x+c in the same context, it is presumed that the discusion is for a particular x.