Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = ⟨cos t + t sin t, sin t - t cos t, t⟩

r(t) = ti + 3tj + t²k, u(t) = 4ti + t²j + t³k Use the properties of the derivative to find the following. (a)r′(t) (b)d/dt [3r(t) - u(t)] (c)d/dt (5t)u(t) (d)d/dt [r(t)·u(t)] (e)d/dt [r(t)×u(t)] (f)d/dt r(2t)

The position vector r describes the path of an object moving in the xy-plane. (c) Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point. Position Vector: r(t) = t² i + tj Point: (4, 2)

Find the following.
(a) $\mathbf{r}^{\prime}(t)$ (b) $\mathbf{r}^{\prime \prime}(t)$ (c ) $D_t[\mathbf{r}(t) \cdot \mathrm{u}(t)]$ (d) $D_t[3 \mathbf{r}(t)-\mathrm{u}(t)]$ (e) $D_t[\mathrm{r}(t) \times \mathrm{u}(t)]$ (f) $D_t[\|\mathrm{r}(t)\|], \quad t>0$
$\mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^2 \mathbf{k}, \quad \mathbf{u}(t)=4 t \mathbf{i}+t^2 \mathbf{j}+t^3 \mathbf{k}$

Find $\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}$, and $a_{\mathrm{N}}$ at the given time $t$ for the plane curve $r(t)$. \

$$\mathbf{r}(t)=\langle\cos \omega t+\omega t \sin \omega t, \sin \omega t-\omega t \cos \omega t\rangle, \quad t=t_0$$

To test the impact speeds of different air supply drops, a pilot is flying horizontally at an altitude of 400 feet with a speed of 60 miles per hour . When should the package be released for it to hit the target? (Give your answer in terms of the angle of depression from the plane to the target.) What is the speed of the package at the time of impact?

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. Position Vector: r(t) = ⟨cos³ t, sin³ t, 3t⟩ Time: t = π

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. $\mathbf{r}(t)=\sqrt{t} \mathbf{i}+5 t \mathbf{j}+2 t^{2} \mathbf{k}$. Time t=4

Find $\frac{d}{d t}[ r (t) \times u (t)]$ in two different ways.

(i) Find the product first, then differentiate.

(ii) Apply the properties of Theorem.

$$r (t)=t i +2 t^2 j +t^3 k , \quad u (t)=t^4 k$$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = √t i + (t² - 1)j + ¼tk

A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector.

The ejector has a fixed angle of $45^{\circ}$. Find the initial speed required.

A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8 feet above and 16 feet behind the ejector.

Find the minimum initial speed of the bale and the corresponding angle at which it must be ejected from the baler.

A projectile is fired from ground level at an angle of 12° with the horizontal. The projectile is to have a range of 200 feet. Find the minimum initial velocity necessary.

Find $\frac{d}{d t}[ r (t) \cdot u (t)]$.

$$r (t)=t i +2 t^2 j +t^3 k , \quad u (t)=t^4 k$$