B Why does the Limit Process give an Exact Slope?

[pbuk]

However, what I don't know is why it is EXACTLY 2x and not just very close to 2x.

Because you can prove it.

1. Start by assuming the opposite: that the slope of the tangent line is not exactly $2x$, it is $2x + \varepsilon$.
2. Now you already know that for $f(x) = x^2$ the slope of the secant line between $x$ and $x + \Delta x$ is $2x + \Delta x$.
3. As you make $\Delta x$ smaller, the slope of the secant line will become closer to the slope of the tangent line - or rather it will never start getting further away (this is the definition of differentiability). So if you believe that the slope of the tangent is actually $2x + \varepsilon$ then the slope of the secant line can never be smaller than this.
4. Choose $\Delta x = \frac \varepsilon 2$. The slope of the secant line is $2 x + \Delta x = 2 x + \frac \varepsilon 2$. From 3. you know that the slope must be greater than or equal to $2 x + \varepsilon$. The only value of $\varepsilon$ for which $2 x + \frac \varepsilon 2 \ge 2 x + \varepsilon$ is $0$.
5. From 1. the slope of the tangent line is $2x + \varepsilon = 2x + 0 = 2x$.

Note that although this proof should be convincing at the 'Basic' level, it has a hole in it and to overcome this hole you need to study calculus more formally which is often called analysis.

@NoahsArk if you can spot the hole and it bothers you then you might find the book Calculus by Michael Spivak interesting. If you prefer learning from a combination of video lectures, notes and exercises then I recommend https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/.

 

[Vanadium 50]

We've posted 32 messages since the OP was here last. Are we helping him? Or just piling on?

 

[pbuk]

We've posted 32 messages since the OP was here last. Are we helping him? Or just piling on?

Yes, there was a big diversion off-topic. I hope that my previous post (which ignores the diversion) is back on-topic and does actually help.

 

[fresh_42]

We've posted 32 messages since the OP was here last. Are we helping him? Or just piling on?

Indeed. We can discuss endlessly about differentiation. Some students once asked me to write a summary about this subject. I ended up with an article that had to be split into five parts to suit the length of typical, however yet long insight articles. I like to say that if you read two authors you will find four notations. There is a long way from a slope to the pullback of sections and it is paved with d's. It cannot be dealt with in a single thread, and by leaving the original limits of the question there is no restriction in place anymore.

I would like to see a discussion about the use of the word infinitesimal, its history, and its meaning in physics, especially as mathematicians have restricted its use to hyperreals and related concepts; but that would require a different thread.

This thread is closed now.