The position vector r describes the path of an object moving in the xy-plane. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. Position Vector: r(t) = ti + (-t² + 4) j Point: (1, 3)

The position vector r describes the path of an object moving in the xy-plane. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = t²i + tj Point: (4, 2)

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

Find the curvature K of the curve at the point P. r(t) = 3ti + 2t²j, P(-3, 2)

Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = t²i + 2tj + 3/2tk

The position vector r describes the path of an object moving in the xy-plane. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = ti + (-t² + 4)j Point: (1, 3)

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) = 4ti + t³j - tk Time: t = 1

A shot fired from a gun with a muzzle velocity of 1200 feet per second is to hit a target 3000 feet away. Determine the minimum angle of elevation of the gun.

Sketch the curve represented by the vector-valued function and give the orientation of the curve.

$$r(t) = 2 sin ti + 2 cos tj + e^t k$$

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) = 4ti + t³j - tk Time: t = 1

The four figures below are graphs of the vector-valued function

$$\mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}+\frac{t}{4} \mathbf{k}$$

Match each of the four graphs with the point in space from which the helix is viewed.

Find the domain of the vector-valued function. r(t) = F(t) x G(t), where F(t) = sin ti + cos tj, G(t) = sin tj + cos tk

Explain why in spherical coordinates the graph of θ = c is a half-plane and not an entire plane.

Analyze the trace when the surface

$$z=\frac{1}{2} x^2+\frac{1}{4} y^2$$

is intersected by the indicated planes.

Find the lengths of the major and minor axes and the coordinates of the foci of the ellipse generated when the surface is intersected by the planes given by $z=8$