Section 1
Vector-Valued Functions
Vector–valued func?ons are closely related to ________ of graphs.
When sketching vector–valued func?ons, technically one isn’t graphing points, but rather ________.
It can be useful to think of __________ as a vector that points from a starting position to an ending position.
In the context of vector–valued functions, average rate of change is ________ divided by tme.
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\left\langle t^{2}, t^{2}-1\right\rangle, \text { for }-2 \leq t \leq 2$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\left\langle t^{2}, t^{3}\right\rangle, \text { for }-2 \leq t \leq 2$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\left\langle 1 / t, 1 / t^{2}\right\rangle, \text { for }-2 \leq t \leq 2$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\left\langle\frac{1}{10} t^{2}, \sin t\right\rangle, \text { for }-2 \pi \leq t \leq 2 \pi$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\langle 3 \sin (\pi t), 2 \cos (\pi t)\rangle, \text { on }[0,2]$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\langle 3 \cos t, 2 \sin (2 t)\rangle, \text { on }[0,2 \pi]$$
Sketch the vector-valued function on the given interval.$$\vec{r}(t)=\langle 2 \sec t, \tan t\rangle, \text { on }[-\pi, \pi]$$
Sketch the vector-valued function on the given interval in $\mathbb{R}^{3}$. Technology may be useful in creating the sketch.$$\vec{r}(t)=\langle 2 \cos t, t, 2 \sin t\rangle, \text { on }[0,2 \pi]$$
Sketch the vector-valued function on the given interval in $\mathbb{R}^{3}$. Technology may be useful in creating the sketch.$$\vec{r}(t)=\langle 3 \cos t, \sin t, t / \pi\rangle \text { on }[0,2 \pi]$$
Sketch the vector-valued function on the given interval in $\mathbb{R}^{3}$. Technology may be useful in creating the sketch.$$\vec{r}(t)=\langle\cos t, \sin t, \sin t\rangle \text { on }[0,2 \pi]$$
Sketch the vector-valued function on the given interval in $\mathbb{R}^{3}$. Technology may be useful in creating the sketch.$$\vec{r}(t)=\langle\cos t, \sin t, \sin (2 t)\rangle \text { on }[0,2 \pi]$$
Find $\|\vec{r}(t)\|$.$$\vec{r}(t)=\left\langle t, t^{2}\right\rangle$$
Find $\|\vec{r}(t)\|$.$$\vec{r}(t)=\langle 5 \cos t, 3 \sin t\rangle$$
Find $\|\vec{r}(t)\|$.$$\vec{r}(t)=\langle 2 \cos t, 2 \sin t, t\rangle$$
Find $\|\vec{r}(t)\|$.$$\vec{r}(t)=\left\langle\cos t, t, t^{2}\right\rangle$$
Create a vector-valued function whose graph matches the given description.A circle of radius 2, centered at $(1,2),$ traced counterclockwise once on $[0,2 \pi]$.
Create a vector-valued function whose graph matches the given description.A circle of radius $3,$ centered at $(5,5),$ traced clockwise once on $[0,2 \pi]$.
Create a vector-valued function whose graph matches the given description.An ellipse, centered at (0,0) with vertical major axis of length 10 and minor axis of length $3,$ traced once counterclockwise on $[0,2 \pi]$
Create a vector-valued function whose graph matches the given description.An ellipse, centered at (3,-2) with horizontal major axis of length 6 and minor axis of length $4,$ traced once clockwise on $[0,2 \pi]$
Create a vector-valued function whose graph matches the given description.A line through (2,3) with a slope of 5 .
Create a vector-valued function whose graph matches the given description.A line through (1,5) with a slope of $-1 / 2$.
Create a vector-valued function whose graph matches the given description.The line through points (1,2,3) and $(4,5,6),$ where $\vec{r}(0)=\langle 1,2,3\rangle$ and $\vec{r}(1)=\langle 4,5,6\rangle$
Create a vector-valued function whose graph matches the given description.The line through points (1,2) and $(4,4),$ where $\vec{r}(0)=\langle 1,2\rangle$ and $\vec{r}(1)=\langle 4,4\rangle$
Create a vector-valued function whose graph matches the given description.A vertically oriented helix with radius of 2 that starts at (2,0,0) and ends at $(2,0,4 \pi)$ after 1 revolution on $[0,2 \pi]$.
Create a vector-valued function whose graph matches the given description.A vertically oriented helix with radius of 3 that starts at (3,0,0) and ends at (3,0,3) after 2 revolutions on [0,1]
Find the average rate of change of $\vec{r}(t)$ on the given interval.$$\vec{r}(t)=\left\langle t, t^{2}\right\rangle \text { on }[-2,2]$$
Find the average rate of change of $\vec{r}(t)$ on the given interval.$$\vec{r}(t)=\langle t, t+\sin t\rangle \text { on }[0,2 \pi]$$
Find the average rate of change of $\vec{r}(t)$ on the given interval.$$\vec{r}(t)=\langle 3 \cos t, 2 \sin t, t\rangle \text { on }[0,2 \pi]$$
Find the average rate of change of $\vec{r}(t)$ on the given interval.$$\vec{r}(t)=\left\langle t, t^{2}, t^{3}\right\rangle \text { on }[-1,3]$$