Atatiliri34
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- Jan 11, 2023
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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).
I believe it is "-x" in A, not "-2x".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 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?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
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.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?
Can you please tell me your definition of a continuous set?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
I just looked it up. Apparently, I'm co-opting a term that means something other than what I had thought. Sorry about that!Can you please tell me your definition of a continuous set?
I thought that "[imath]\cap[/imath]" stands for the intersection not the union.What I meant was that we may break the union of the sets into "small" segments
How is it that I can talk about Topology and yet miss that??I thought that "[imath]\cap[/imath]" stands for the intersection not the union.