Expert Assistance for Parametric equations: Comprehensive Resources and Practice Problems

Find the point(s) of intersection of the following two parametric curves, by first eliminating the parameter, then solving the system of equations.
${\displaystyle x=t+5,y=t^2\text{and}x=\frac{1}{16}t,y=t}$
a. (25, 400)
b. (1, 16)
c. (400, 16)
d. A and B
e. A and C

Find ${\displaystyle \frac{d^{3}y}{{dx}^3}}$ for the following parametric equation:
$x=t^2-3$
$y=t^8$
$t=2$

Convert the point $(-1,1)$ from Cartesian coordinates to polar coordinates

Convert each polar equation to rectangular form. ${\displaystyle =-\frac{\pi }{6}}$

Consider the parametric equation ${\displaystyle x=2t-1,y=8t^3.-1\le t\le 1}$

a)Eliminate the parameter to obtain an equation in x and y.

b)Sketch the graph of the equation

Represent the line segment from P to Q by a vector-valued function
$P(-8,-2,-2),Q(-3,-9,-4)$
Represent the line segment from P to Q by a set of parametric equations

Change from rectangular to spherical coordinates. (Let ${\displaystyle \rho \ge 0,0\le \theta \le 2\pi ,\text{ and }0\le \varphi \le \pi }$)
(a) $(0,-3,0)$
(b) ${\displaystyle (-1,1,-\sqrt{2})}$

Find the parameric equatins to represent the line segment from $(1,2)$ to $(3,5)$. Give the domain of the parameter