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.) (b) θ = 10°, v_0 = 146 ft/sec

Sketch the plane curve and find its length over the given interval. r(t) = (t + 1)i + t²j, [0, 6]

Find a set of parametric equations for the tangent line to the graph of the vector-valued function at $t=t_\theta$. Use the equations for the line to approximate $r\left(t_0+0.1\right)$.
$\mathbf{r}(t)=3 \cosh t \mathbf{i}+\sinh t \mathbf{j}-2 t \mathbf{k}, \quad t_0=0$

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

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

Find the curvature K of the curve at the point P. r(t) = e^t i + 4tj, P(1, 0)

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

$$\mathbf{r}(t)=\langle\omega t-\sin \omega t, 1-\cos \omega t\rangle, \quad t=t_0$$

Consider an object moving according to the position vector r(t) = a cos ωt i + a sin ωt j. Determine the speed of the object at any time t and explain its value relative to the value of a_T.

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 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)

A projectile is fired from ground level at an angle of 12 degree with the horizontal. The projectile is to have a range of 150 feet. Find the minimum initial velocity necessary.

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

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⟩

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