@fresh_42 These ideas are very subtle and I hope one day to have a better handle on them and to be more proficient in understanding the development of math ideas. I believe this is a difficult task given the long history.
Thanks @fresh_42. I will check out Dieudonne.
I was thinking yesterday that there must have been equations representing relationships between quantities long before Descartes. What seems part of the novelty is having equations for shapes/geometrical objects like lines, circles, etc. Without an...
For a while I've been trying to get a better understanding of how Descartes' invention of Cartesian space revolutionized math. It seems like an invention on par with the invention of agriculture in how it led to human progress. I am still having trouble, though, pinpointing examples of what can...
@PeroK I remember asking about the chain rule and implicit differentiation, but not about u substitution. If I did ask about it and you have a link to the prior discussion, please paste it. Thanks
Can someone please give as simple an example as possible to show what U substitution is about? I know basic integration rules but don't understand the point of u-substitution. I've read that it's used to "undo the chain rule", but I don't see how, and don't see how we can spot when we'd need to...
@Ibix Ok thanks for clarifying. While I am aware that ## \frac {f(x + \Delta x)} {\Delta x} = 2x + \Delta x ##, this explanation has still been not totally convincing for me because the two slope functions aren't really exactly the same. One of them, ## \frac {f(x + \Delta x)} {\Delta x} ## has...
@Ibix I know the answer to this is 2x both from doing the limit process and also from learning the power rule. I also have an intuitive visual sense of why this is the case. However, what I don't know is why it is EXACTLY 2x and not just very close to 2x. Seems like a Xeno's paradox type issue.
@ PeroK, I meant you you get the slope of the tangent line by taking the limit as ##\Delta x## approaches zero of the slope function ## \frac {f(x +h) - f(x)} {h}## (where h = ##\Delta x##). My confusion lies in the fact that although very small changes in x will give closer and closer...
@fresh_42 yes, my confusion is not in what a tangent line is- I'm confused about how we know the value of the slope of the tangent line from using the limit process. I am confused because the process is setting ##\Delta x ## to be zero, but you can't have a zero in the denominator. So, while we...
Thank you for the response @PeroK . I know that the slope is a single number. My question is how do we know for sure what the exact number is. We can never set ##\Delta x## to zero to know an exact number for sure.
I am taking a summer calculus class now. For years I've been stuck on the question of why the limit process gives us an exact slope of the tangent line instead of just a very close approximation. I don't need to know the reason for this class I'm taking- we are basically just learning rules of...