Need explanation to the answers

[Atatiliri34]

Hey,

I already have the fact sheet for this question, but it only shows the answer to each sub question. Could someone show me the explanations?

All the best

 

[stapel]

It is difficult to read the tiny attachment. I think the exercise is as follows:

3. Let A and B be the following sets:
\(\displaystyle A\ =\ \{(x,\ y)\ \in\ \mathbb{R}^2\ \ |\ \ y^2-x+2y+1=0\ \}\)
\(\displaystyle B\ =\ \{(x,\ y)\ \in\ \mathbb{R}^2\ \ |\ \ \frac{x^2}{4}+y^2=1\ \}\)

a. Represent A and B on the Cartesian plane.

Consider the following set of lines:
\(\displaystyle L_k\ =\ \{(x,\ y)\ \in\ \mathfrak{R}^2\ \ |\ \ y=kx\ \}\)
...with parameter \(\displaystyle k\ \in\ \mathbb{R}\).

b. Find the point(s) of the set B where its tangent line is orthogonal to \(\displaystyle L_k\) in the case \(\displaystyle k=0\).

c. Find the point(s) of the set A where its tangent line is parallel to \(\displaystyle L_k\) in the case \(\displaystyle k=\frac{1}{2}\).

Let's now consider the subset \(\displaystyle A\cap B\).

d. Find the coordinates of the singular point(s) of the curve representing \(\displaystyle A\cap B\) (that is, the points where the curve is not differentiable) which have a negative ordinate.

e. Compute the angle(s) formed between the tangent lines at this/these point(s).

Please reply with corrections, if necessary. Either way, please reply with a clear listing of your thoughts and efforts so far.

Thank you!

Eliz.

 

[blamocur]

It is difficult to read the tiny attachment. I think the exercise is as follows:



Please reply with corrections, if necessary. Either way, please reply with a clear listing of your thoughts and efforts so far.

Thank you!

Eliz.

I believe it is "-x" in A, not "-2x".

 

[blamocur]

Hey,

I already have the fact sheet for this question, but it only shows the answer to each sub question. Could someone show me the explanations?

All the best

I am confused by "curve representing [imath]A\cap B[/imath]" : I believe [imath]A\cap B[/imath] is a finite set of points, not a curve. Am I missing something here?

 

[topsquark]

I am confused by "curve representing [imath]A\cap B[/imath]" : I believe [imath]A\cap B[/imath] is a finite set of points, not a curve. Am I missing something here?

Set A is a parabola and set B is a circle. They intersect so there is a "curve." (But I'm not sure I'd call it that.) Anyway, there is a continuous set of points involved.

-Dan

 

[Steven G]

Set A is a parabola and set B is a circle. They intersect so there is a "curve." (But I'm not sure I'd call it that.) Anyway, there is a continuous set of points involved.

-Dan

Can you please tell me your definition of a continuous set?

 

[topsquark]

Can you please tell me your definition of a continuous set?

I just looked it up. Apparently, I'm co-opting a term that means something other than what I had thought. Sorry about that!

What I meant was that we may break the union of the sets into "small" segments and define a continuous mapping that takes us along the curve. (Something like local path continuity.) Technically speaking we actually have two of these, but since the sets intersect, I had presumed that we could still write a continuous function to get from any one point in the set to another by a continuous path.

Perhaps my Topology is off?

-Dan

 

[blamocur]

What I meant was that we may break the union of the sets into "small" segments

I thought that "[imath]\cap[/imath]" stands for the intersection not the union.

 

[topsquark]

I thought that "[imath]\cap[/imath]" stands for the intersection not the union.

How is it that I can talk about Topology and yet miss that??

I'm sick. Leave me alone!

-Dan