Study guide and
5 practice problems
on:
Parameterized curves
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Study Guide
The circle of radius $r$, traversed counterclockwise, can be parameterized by $$\bfx(\theta) = \langle r \cos \theta, r \sin \theta \rangle \text{ for } 0 \leq \theta \lt 2 \pi$$
(3 problems)
To parameterize a curve consisting of multiple types of motion (e.g. translation + rotation):
Describe the desired point as the vector sum of each part
Identify a convenient parameter. When in doubt, choose time
Find each part's dependence on the parameter
(2 problems)
Velocity of a parameterization: $\bfv = \frac{d \bfx} {dt}$
(3 problems)
Speed of a parameterization: $|\bfv| = \left| \frac{d\bfx}{dt} \right |$
(3 problems)
The velocity vector $\frac{d \bfx}{dt}$ is tangent to the curve $\bfx(t)$.
(0 problems)
The graph of the function $y = f(x)$ can be parameterized by $x=t$ and $y = f(t)$.
(1 problem)
The length of the curve $\bfx(t)$ traced out between $t=a$ and $t=b$ is given by $$s = \int_a^b \left | \frac{d \bfx(t)}{dt} \right| dt.$$
(2 problems)
Related topics
Multivariable calculus
(147 problems)
Lines, Planes, and Curves
(13 problems)
Practice problems
Consider a cylinder of radius $r$ rolling up a hill of incline $\theta$ at constant speed $v$. Initially the point of contact is $(0,0)$. Find the trajectory of the point initially contacting the hill.
Solution
A frisbee of radius $r$ translates rightward at speed $v$ meter/second. It rotates clockwise at $\omega$ radian/second. Initially the frisbee is centered at the origin.
Find the trajectory swept by the point initially at $(0,r)$.
Compute the speed as a function of time.
Describe when the speed is largest and smallest.
Solution
Find the arc length of the helix $x(t) = \cos t, y(t) = \sin t, z(t) = t$ traced from $t=1$ to $t=2$.
Solution
An ant is on a merry-go-ground that is rotating clockwise at $\omega$ radians per second. Initially, the ant is at $(R,0)$. From the ant's perspective, it walks toward the center with speed $v$. Several snapshots in time are as follows:
Find the parameterization of the path taken by the ant (relative to the ground)
Compute the speed of the ant as a function of $t$. When is it largest?
Set up, but do not evaluate, an integral for the arc length of the path taken by the ant between $t=0$ and when the ant reaches the origin
Solution
Directly compute $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F}(x,y) = x y \ \mathbf{i} + \mathbf{j}$ and $C$ is the curve connecting $(0,0)$ to $(1,1)$ along $y=x^2$.
Solution