Help understanding parametric curve problems

[thatguy47]

For each problem:
a)Eliminate the parameter to find a Cartesian equation of the curve.
b)Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.

13.
Problem:
x= sin^2(theta) -->>also written as (sin(theta))^2
y = cos^2(theta) ....aka y= (cos(theta))^2

Answer:

I understand up to

. Can someone please explain the second half of the answer to me?

15)
Problem:
x=e^t
y=e^(-t)

Answer:

How do you know that x>0 ?

 

[Deleted member 4993]

Can someone please explain the second half of the answer to me?

What exactly you don't understand in the second half. Better yet can you please re-write the second half - because part of it has been cut-off.

15)
Problem:
x=e^t
y=e^(-t)

Answer:

How do you know that x>0 ?

The range of exponential function (e[sup:268nr5eu]t[/sup:268nr5eu]) is positive. In other words, e[sup:268nr5eu]t[/sup:268nr5eu] cannot become negative for any value of t.

 

[thatguy47]

Thanks for explaining the second problem. I don't understand on #13 part a: how do you get the interval 0</x</1 ?

 

[stapel]

The text (cut off in the image as displayed in the forum) is as follows:

\(\displaystyle \mbox{13. (a) } x\, =\, \sin^2(\theta),\, y\, =\, \cos^2(\theta),\)
\(\displaystyle x\, +\, y\, =\, \sin^2(\theta)\, +\, \cos^2(\theta)\, =\, 1,\, 0\, \leq\, x\, \leq\, 1\)

\(\displaystyle \mbox{Note that the curve is at }\, (0,\, 1)\, \mbox{whenever}\theta\, =\, \pi n\, \mbox{ and}\)
\(\displaystyle \mbox{is at }\, (1,\, 0)\, \mbox{whenever }\, \theta\, =\, \frac{\pi}{2}n\, \mbox{for every integer }\, n.\)