Find where the tangent line in the first quadrant is horizontal for the Folium of Descartes

[Callumnc1]

For this problem,

I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation?

Many thanks!

 

[Mark44]

For this problem,
View attachment 322953
I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables. Would anybody else be able to provide with an explanation?

Many thanks!

The "bottom fraction", really the bottom of the fraction, or denominator, can't be zero because division by zero is undefined. So the subsequent work is to determine the values of the x and y variables that make the fraction, and hence y', undefined.

This is very basic stuff.

 

[Callumnc1]

The "bottom fraction", really the bottom of the fraction, or denominator, can't be zero because division by zero is undefined. So the subsequent work is to determine the values of the x and y variables that make the fraction, and hence y', undefined.

This is very basic stuff.

Thank you for your help @Mark44 !

 

[Mark44]

I don’t understand why the bottom fraction cannot equal zero since we are not plugging in any numbers it just in terms of variables.

It's really immaterial whether we're dealing with constants or variables, and in the latter case, whether we're "plugging in numbers" or not. There are certain arithmetic and algebraic operations that aren't allowed, among which the most prominent is division by zero. Other operations that aren't allowed, if the result is intended to be a real number, are taking the square root or other even root of a negative number, taking the logarithm or a non-positive number, and so on.

These ideas normally are presented before derivatives are presented, but it looks like you might have skipped over or forgotten them.

 

[Callumnc1]

It's really immaterial whether we're dealing with constants or variables, and in the latter case, whether we're "plugging in numbers" or not. There are certain arithmetic and algebraic operations that aren't allowed, among which the most prominent is division by zero. Other operations that aren't allowed, if the result is intended to be a real number, are taking the square root or other even root of a negative number, taking the logarithm or a non-positive number, and so on.

These ideas normally are presented before derivatives are presented, but it looks like you might have skipped over or forgotten them.

Thank you for your reply @Mark44!

They did not really teach me too much about those ideas sadly. I got taught cacl before conics and alot of the other precacl topics.

Many thanks!

 

[Mark44]

They did not really teach me too much about those ideas sadly. I got taught cacl before conics and alot of the other precacl topics.

Then it would be a very good idea to spend some time going over the precalculus topics, either from a precalc textbook (which you can get fairly cheaply) or via internet sites such as khanacademy.org. Not having a solid foundation in algebra and trig is really going to hold back your understanding in the kinds of problems you've been posting.

 

[Callumnc1]

Then it would be a very good idea to spend some time going over the precalculus topics, either from a precalc textbook (which you can get fairly cheaply) or via internet sites such as khanacademy.org. Not having a solid foundation in algebra and trig is really going to hold back your understanding in the kinds of problems you've been posting.

Thank you @Mark44, I will have a look!