The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of Position Vector: r(t) = ⟨cos³ t, sin³ t, 3t⟩ Time: t = π

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)

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⟩

Find T(t), N(t), a_T, and a_N at the given time t for the space curve r(t). Vector-Valued Function: r(t) = cos ti + sin tj + 2tk Time: t = π/3

Use a graphing utility to graph the paths of a projectile for the given values of θ and v_0. For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.) (a) θ = 10°, v_0 = 66 ft/sec

The indicated figure shows the path of a particle modeled by the vector-valued function

$$r(t)=\langle\pi t-\sin \pi t, 1-\cos \pi t\rangle .$$

The figure also shows the vectors $v(t) /\|v(t)\|$ and $a(t) /\|a(t)\|$ at the indicated values of t.

Find $a_{T}$ and $a_{N}$ at $t=\frac{1}{2}, t=1$, and $t=\frac{3}{2}$.

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

In the following exercise, consider the vector-valued function $r(t)=3 t^2 i+(t-1) j+t k$. Write a vector-valued function u(t) that is the specified transformation of r.

The y-value increases by a factor of two

In the following exercise, find the indefinite integral.

$$\int\left(e^t i+j+t \cos t k\right) d t$$

In the following exercise, find the curvature and radius of curvature of the plane curve at the given value of x.

$$y=\sin 2 x, \quad x=\frac{\pi}{4}$$

Find the indefinite integral. $\int(2 t \mathbf{i}+\mathbf{j}+9 \mathbf{k}) d t$

The path of a shot thrown at an angle $\theta$ is $r(t)=\left(v_0 \cos \theta\right) t i+\left[h+\left(v_0 \sin \theta\right) t-\frac{1}{2} g t^2\right] j$ where $v_0$ is the initial speed, h is the initial height, t is the time in seconds, and g is the acceleration due to gravity. Verify that the shot will remain in the air for a total of $t=\frac{v_0 \sin \theta+\sqrt{v_0^2 \sin ^2 \theta+2 g h}}{g}$ seconds and will travel a horizontal distance of $\frac{v_0^2 \cos \theta}{g}\left(\sin \theta+\sqrt{\sin ^2 \theta+\frac{2 g h}{v_0^2}}\right)$ feet.

The position vector r describes the path of an object moving in space. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector: r(t) = √t i + 5tj + 2t²k Time: t = 4

Graph the vector-valued function and identify the common curve. r(t) = ½t²i + tj - √3/2 t²k