Vector-Valued Functions of One Variable

The same curve can be represented with different parameterizations. Use appropriate technology to plot each of the curves generated by the following vector-valued functions for values of $t$ from $0$ to $2 \pi$. For each example, you should write a few sentences to explain how the graphs are alike and how they are different. Be sure to pay attention to how the parameter values are related to different point…

For this activity we will consider the paraboloid with equation $z = x^2+y^2$. You do not need to know what the graph of $z=x^2+y^2$ looks like at this point.

1. Parameterize the intersection of the paraboloid with the fundamental plane $x=2$. What type of curve is the intersection?

2. Parameterize the intersection of the paraboloid with the fundamental plane $y = -1$. What type of curve is the intersection?

3. Parameterize the intersection…

A standard parameterization for the unit circle is $\langle \cos(t), \sin(t) \rangle$, for $0 \le t \le 2\pi$.

1. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ and travels clockwise along the circle as $t$ increases.

2. Find a vector-valued function $\vr$ that describes a point traveling along the unit circle so that at time $t=0$ the point is at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$ and travels counter-clockwise along the…

You might recall the pythagorean identity from trigonometry given by We want to use this identity to parameterize the circle with center $(h,k)$ and radius $r \gt 0$ given by

1. Reorganize the general equation of a circle to be of the form

2. Because identities work for any value, we can use the pythagorean identity to parameterize the general equation of a circle. Fill in the blanks below with each of the respecti…

Let $a$ and $b$ be positive real numbers. You have probably seen the equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ that generates an ellipse, centered at $(h,k)$, with a horizontal axis of length $2a$ and a vertical axis of length $2b$.

1. Write a few sentences to explain why the vector function $\vr$ defined by $\vr(t) = \langle a\cos(t), b\sin(t) \rangle$, $0 \le t \le 2\pi$ is a parameterization of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

2. Find a parameterization of the ellipse $\frac{x^2}{4} + \frac{y^2}{16} = 1$ that is traversed clockwise.

3. Find a parameterization of the e…

In this exercise, we will examine the graph of $z = 3x^2 + 4y^2 - 2$. You do not need to know what this graph looks like, but we will look at the intersection of this graph with several types of fundamental planes.

1. Determine a vector-valued function $\vr$ that parameterizes the curve which is the $x = 2$ trace of $z = 3x^2 + 4y^2 - 2$. Plot the resulting curve. Do likewise for $x = -2, -1, 0,$ and $1$.

2. Determine a vector-valued function $\vr$ that parameterizes the cur…

Recall that any line in space may be represented parametrically by a vector-valued function.

1. Find a vector-valued function $\vr$ that parameterizes the line through $(-2,1,4)$ in the direction of the vector $\vv = \langle 3, 2, -5 \rangle$.

2. Find a vector-valued function $\vr$ that parameterizes the line of intersection of the planes $x + 2y - z = 4$ and $3x + y - 2z = 1$.

3. Determine the point of intersection of the lines given by Then, find a vector-valued function $\vr$ that para…

For each of the following, describe the effect of the parameter $s$ on the parametric curve for $t$ in the interval $[0,2 \pi]$.

1. $\vr(t) = \langle \cos(t), \sin(t) + s \rangle$

2. $\vr(t) = \langle \cos(t)-s, \sin(t) \rangle$

3. $\vr(t) = \langle s\cos(t), \sin(t) \rangle$

4. $\vr(t) = \langle s\cos(t), s\sin(t) \rangle$

5. $\vr(t) = \langle \cos(st), \sin(st) \rangle$

For each of the following equations, give a parameterization and be sure state the bounds of your parameterization.

1. $\frac{(x-2)^2}{9}+\frac{(y+1)^2}{16}=1$

2. $\frac{(y+3)^2}{4}-\frac{(x+1)^2}{9}=1$

3. $(y-7)^2=-4(x+1)$