Represent the plane curve by a vectorvalued function. (There are many correct answers.) 3x + 4y - 12 = 0

In the following exercise, find the principal unit normal vector to the curve at the specified value of the parameter.

$$r(t)=6 \cos t i+6 \sin t i+k, \quad t=\frac{3 \pi}{4}$$

Consider the helix represented by the vector-valued function $r (t)=\langle 2 \cos t, 2 \sin t, t\rangle$.

Verify that $\left\| r ^{\prime}(s)\right\|=1$.

Find r′(t). r(t) = 6ti - 7t²j + t³k

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. P(0, 2, -1), Q(4, 7, 2)

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = √t i + t^3/2 j + e^-t/4 k (d) r(9 + Δt) - r(9)

Consider the helix represented by the vector-valued function $r (t)=\langle 2 \cos t, 2 \sin t, t\rangle$.

Find the coordinates of the point on the helix for arc lengths $s=\sqrt{5}$ and $s=4$.

The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector. $\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}$. Time t=4

Find $\mathbf{r}^{\prime}(t)$. $\mathbf{r}(t)=3 \cos ^{3} t \mathbf{i}+2 \sin ^{3} t \mathbf{j}+\mathbf{k}$

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. $P(1,-6,8), Q(-3,-2,5)$

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = √t i + t^3/2 j + e^-t/4 k (c) r(c + 2)

Consider the vector-valued function $\mathbf{r}(t)=\langle t \cos \pi t, t \sin \pi t\rangle$, $0 \leq t \leq 2$. (a) Use a graphing utility to graph the function. (b) Find the length of the arc in part (a). (c) Find the curvature K as a function of t. Find the curvature at t=0, t=1, and t=2. (d) Use a graphing utility to graph the function K. (e) Find (if possible) $\lim _{t \rightarrow \infty} K$. (f) Using the result of part (e), make a conjecture about the graph of r as $t \rightarrow \infty$.

Consider the helix represented by the vector-valued function $r (t)=\langle 2 \cos t, 2 \sin t, t\rangle$.

Solve for t in the relationship derived in part (a), and substitute the result into the original vector-valued function. This yields a parametrization of the curve in terms of the arc length parameter $s$.

In the following exercise, the position vector r describes the path of an object moving in space.

Position Vector: $r(t)=t^2 i+t j+2 t^{3 / 2} k$

Time: t = 4

Evaluate the velocity vector and acceleration vector of the object at the given value of t.