Question

Find a parametric representation c(t) of a path that travels in the increasing y-direction along the graph of x + y^2/25=1 in R2, for y≥0. Further require that c(0) = (1,0) and c(5)=(0,5).

Answer 1

r=3+sin(θ)
Let's consider r=3+sin(θ). Since the values of sine are all between –1 and 1, r will be between 2 and 4. Any points on this curve will have distance to the origin between 2 and 4 (the green and red circles on the accompanying graph). When θ=0 (the positive x-axis) r is 3. As θ increases in a counterclockwise fashion, the value of r increases to 4 in the first quadrant. In the second quadrant, r decreases from 4 to 3. In the third quadrant, corresponding the sine's behavior (decrease from 0 to –1) r decreases from 3 to 2. In all of this {in|de}crease discussion, the geometric effect is that the distance to the origin changes. We're in a situation where the central orientation is what matters, not up or down or left or right. Finally, in the fourth quadrant r increases from 2 to 3, and since sine is periodic with period 2Π, the curve joins its earlier points.
The picture to the right shows the curve in black. I'd describe the curve as a slightly flattened circle. The flattening is barely apparent to the eye, but if you examine the numbers, the up/down diameter of the curve is 6, and the left/right diameter is 6.4.

Answer 2

Is this the right problem?

Answer 3

Yes, Believe Me

Answer 4

Where is the picture?

Answer 5

I Cannot Make The Picture ? Tahts Very Difficult

Answer 6

Then why'd you say the picture on the right if I can't see it

Answer 7

Wait THe Minute I Make IT

Answer 8

\[c(t)=1-\frac{ t^2 }{ 25 }\]