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 (b) r(4)

Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = 2 cos ti + tj + 2 sin tk

Sketch the space curve and find its length over the given interval. Vector-Valued Function: r(t) = ⟨2 sin t, 5t, 2 cos t⟩

(c) Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point. Position vector: r(t) = 3ti + (t-1)j Point: (3, 0)

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) = ti + t²j + ½t²k Time: t = 4

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = ln ti + 1/t j + 3tk (d)r(1+Δt) -r(1)

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = ln ti + 1/t j + 3tk (c)r(t-4))

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) = ti + t²j + ½t²k Time: t = 4

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = ln ti + 1/t j + 3tk (b)r(-3)

Prove that dN/ds = -KT + τB.

Evaluate (if possible) the vector-valued function at each given value of t. r(t) = ln ti + 1/t j + 3tk (a)r(2)

Find r′(t). r(t) = √t i + (1 - t³)j

Prove that there exists a scalar τ, called the torsion, such that dB/ds = -τN.

Sketch the space curve and find its length over the given interval. Vector-Valued Function: r(t) = ⟨4t, -cos t, sin t⟩ Interval: [0, 3π/2]