Vector Functions

Find 

(a) the displacement vectors from r(a) tor(b), 

(b) the magnitude of the displacement vector, and

 (c) the distance travelled by a particle on the curve from a to b.

 r(t) = t, t² a = 2, b = 3

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

Parametric equations for the unit circle: Find parametric equations for the unit circle centered at the origin of the xy-plane that satisfy the given conditions.

The graph is traced counterclockwise once on the interval [0, 2π] starting at the point (1, 0).

(a) Parametric equations with three components for a

circular helix winding around the $x-$axis

Parametric equations for the unit circle: Find parametric equations for the unit circle centered at the origin of the xy-plane that satisfy the given conditions.

The graph is traced clockwise once on the interval [0, 2π] starting at the point (1, 0).

Let $y = f\left(x\right)$. What is the definition of $\underset{x\to c}{\lim }f\left(x\right) = L$?

Parametric equations for the unit circle: Find parametric equations for the unit circle centered at the origin of the xy-plane that satisfy the given conditions.

The graph is traced counterclockwise twice on the interval [0, 2π] starting at the point (1, 0).

Parametric equations for a circle: Find parametric equations whose graph is the circle with radius ρ centered at the point (a, b) in the xy-plane such that the graph is traced counterclockwise k > 0 times on the interval [0, 2π] starting at the point (a + ρ, b).

Explain why we do not need an “epsilon–delta” definition for the limit of a vector-valued function.

Let y = f(x). State the definition for the continuity of the function f at a point c in the domain of f . 

Let y = f (x). State the definition for the continuity of the function f at a point c in the domain of f .

Most of the parametric equations and vector-valued functions we have studied have component functions that are continuous. What happens when one of the component functions is discontinuous at a point? For example, the “floor” function $z\left(t\right)=⌊t⌋$ has a jump discontinuity for every integer $t$. What is the graph of the equations $x = \cos 2\pi t, y = \sin 2\pi t,z = t, t \in R$?

Let $\mathit{r}\left(t\right) = x\left(t\right), y\left(t\right), t \in [a, b],$ be a vector-valued function, where $a < b$ are real numbers and the functions $x\left(t\right)$ and $y\left(t\right)$are continuous. Explain why the graph of $\mathit{r}$ is contained in some circle centered at the origin. (Hint: Think about the Extreme Value Theorem.)

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right), t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in a circle centered at the origin. (Hint: Graph the functions $r_1\left(t\right)=\left(\frac{1}{t},\frac{1}{t}\right)$ and $r_2\left(t\right)=\left(t,t\right),$ both with domain [1,∞).)

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right),z\left(t\right)\right), t\in [a, b],$ be a vector-valued function, where a < b are real numbers and the functions x(t), y(t), and z(t) are continuous. Explain why the graph of r is contained in some sphere centered at the origin.

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right), z\left(t\right)\right), t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Explain why the graph of r may or may not be contained in some sphere centered at the origin. (Hint: Consider the functions $r_1\left(t\right)=\left(\cos t, \sin t, 1/t\right)$ and $r_2\left(t\right)=\left(\cos t, \sin t, t\right), \mathrm{both} \mathrm{with} \mathrm{domain} [1,\infty ).)$

As we saw in Example 1, the graph of the vector-valued function $r\left(t\right)=\left(\cos t, \sin t, t\right), \mathrm{for} t\in [0, 2\pi ]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

As we saw in Example 1, the graph of the vector-valued function $r\left(t\right)=\left(\cos t, \sin t, t\right), \mathrm{for} t\in [0, 2\pi ]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right), t\in [a,\infty ),$be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as $t\to \infty ?$ Provide an example illustrating your answer.

Let $r\left(t\right)=\left( x\left(t\right), y\left(t\right)\right), t\in [a,\infty ),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t → ∞? Provide an example illustrating your answer.

What is the dot product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right)\right) \mathrm{and} r_2\left(t\right)=\left(x_2\left(t\right), y_2\left(t\right)\right)?$

Compute the cross product of the vector functions $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right)\right) \mathrm{and} \left(r_2\left(t\right)=x_2\left(t\right), y_2\left(t\right)\right)$ by thinking of ${\mathrm{ℝ}}^2$ as the xy-plane in ${\mathrm{ℝ}}^3.$ That is, let $r_1\left(t\right)=\left(x_1\left(t\right), y_1\left(t\right), 0\right) \mathrm{and} r_2\left(t\right)=\left(x_2\left(t\right), y_2\left(t\right), 0\right),$ and take the cross product of these vector functions.

In Exercises 19–21 sketch the graph of a vector-valued function $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right)$ with the specified properties. Be sure to indicate the direction of increasing values of t.

Domain $t\ge 0, r\left(0\right)=\left(0,3\right), r\left(1\right)=\left(-2,1\right), \underset{t\to \infty }{\lim }x\left(t\right)=-5, \mathrm{and} \underset{\mathrm{t}\to \infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty .$

In Exercises 19–21 sketch the graph of a vector-valued function $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right)$ with the specified properties. Be sure to indicate the direction of increasing values of t.

Domain $t\ge 0, r\left(0\right)=\left(1,0\right), r\left(2\right)=\left(0,-1\right), \underset{t\to \infty }{\lim }x\left(t\right)=-2, \mathrm{and} \underset{\mathrm{t}\to \infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-3.$

In Exercises 19–21 sketch the graph of a vector-valued function $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right)$ with the specified properties. Be sure to indicate the direction of increasing values of t.

Domain

 $$\begin{gathered} t\in \mathrm{ℝ}, r\left(2\right)=\left(1,1\right), r\left(0\right)=\left(-3,3\right), r\left(-2\right)=\left(-5,-5\right) \underset{t\to -\infty }{\lim }x\left(t\right)=\infty , \\ \underset{\mathrm{t}\to -\infty }{\lim }\mathrm{y}\left(\mathrm{t}\right)=-\infty , \mathrm{and} \underset{t\to \infty }{\lim }\mathrm{r}\left(\mathrm{t}\right)=\left(0,0\right). \end{gathered}$$

Given a vector-valued function r(t) with domain $\mathrm{ℝ},$ what is the relationship between the graph of r(t) and the graph of kr(t), where k > 1 is a scalar?

Given a vector-valued function r(t) with domain $\mathrm{ℝ},$ what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

Explain why the graph of every vector-valued function $r\left(t\right)=\left(\cos t, \sin t, f\left(t\right)\right)$lies on the surface of the cylinder $x^2+y^2=1$ for every continuous function f.

Explain why the graph of every vector-valued function $r\left(t\right)=\left(\cos t, \sin t, \cos t\right)$ lies on the intersection of the two cylinders $x^2+y^2=1 \mathrm{and} y^2+z^2=1.$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(\sin 3t, \cos 3t) for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(2-\sin t, 4+\cos t) for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(\sin t, \cos 2t) for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(1+\sin t, 3-\cos 2t), for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(3+t)i+(3-\frac{1}{t})j for t>0$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(t,t^2,t^3) for t\in [0,2]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=({\cos }^{2}t, 4int, t) for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=({\cos }^{2}t, \sin 2t) for t\in [0,2\mathrm{\pi }]$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r\left(t\right)=(t \sin t, t \cos t, t) for t\in [0,4\mathrm{\pi }]$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$(1,3t,t^3)+(t,t^2,t^3)$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$(1,t,t^2)-(t,t^2,t^3)$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$5(\cos t, \sin t)$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$t(\sin t, \cos t)$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$(\sin t, \cos t).(\cos t, -\sin t)$

Evaluate and simplify the indicated quantities in Exercises 35–41.
$\left(\right(\sin t)i+(\cos t)j+tk)\times \left(\right(\cos t)i+(\sin t\left)j\right)$

Evaluate and simplify the indicated quantities in Exercises 35–41.

$(1,t,t^2)\times (1,t^2,t^3)$

Evaluate the limits in Exercises 42–45.

$\underset{t\to 0}{lim}(\sin 3t, \cos 3t)$

Evaluate the limits in Exercises 42–45.

$\underset{t\to \mathrm{\pi }}{lim}(\sin t, \cos t, sect)$

Evaluate the limits in Exercises 42–45.

$\underset{t\to 1^{-}}{lim}(\ln t, \frac{e^t-1}{t-1},e^t)$