Slopes of tangent lines of parametric curves.

[rmiller70015]

1. The problem statement,ll variables and given/known data
I have the first and second derivatives of a parametric function and the book is asking for when the slope of the tangent is vertical and horizontal. I get that horizontal is when dy/dx is 0. But what about vertical, is that dy/dx is 1?

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

1. The problem statement,ll variables and given/known data
I have the first and second derivatives of a parametric function and the book is asking for when the slope of the tangent is vertical and horizontal. I get that horizontal is when dy/dx is 0. But what about vertical, is that dy/dx is 1?

If the tangent line has slope 1, wouldn't it be at a 45 degree angle? Think about dx/dy at a vertical tangent line point.

 

[Mentallic]

When it's vertical, dy/dx is undefined. This is an abuse of notation, but it may help you to think of an undefined value as being $1/0 = \pm \infty$ So if you have

$$\frac{dy}{dx}=\frac{f(x)}{g(x)}$$

Then to find where the tangent is horizontal, you need to evaluate f(x)=0 and to find where it is vertical you calculate g(x)=0. But keep in mind that when solving either one, the x value you find cannot also be a zero of the other. So for example,

$$y=log(x-1)$$ x>1

$$\frac{dy}{dx}=\frac{1}{x-1}=\frac{x}{x(x-1)}$$ I just added a factor of x into the numerator and denominator for illustrative purposes. It doesn't change the function over real values of x.

There is a vertical tangent to the function y at x=1, but at x=0 there is neither a vertical or horizontal even though x=0 gives us 0 in the numerator and in the denominator.

 

[rmiller70015]

Thanks so very very much for this clears up a lot.