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) = 3ti + tj + ¼t²k Time: t = 2

Find r(t) · u(t). Is the result a vector-valued function? Explain. r(t) = (3t - 1)i + ¼t³j + 4k, u(t) = t²i - 8j + t³k

Find the conservative vector field for the potential function by finding its gradient. g(x, y, z) = y/z + z/x - xz/y

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)

Represent the plane curve by a vectorvalued function. (There are many correct answers.) 3x + 4y - 12 = 0

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 (a) r(0)

Determine whether r(t) is continuous at t=0. Explain your reasoning.
r(t)=3 sin ti-2tj

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) = 3ti + tj + ¼t²k Time: t = 2

Solve using Lagrange multipliers. Find a vector in 3-space whose length is 5 and whose components have the largest possible sum.

Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P. r(t) = ⟨2 sin t, 2 cos t, 4 sin² t⟩ P(1, √3, 1)

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

Find r′(t). r(t) = ⟨t cos t, -2 sin t⟩

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)

Let Q(x, y) be the statement “x has sent an e-mail message to y,” where the domain for both x and y consists of all students in your class. Express each of these quantifications in English. a) ∃x∃yQ(x, y) b) ∃x∀yQ(x, y) c) ∀x∃yQ(x, y) d) ∃y∀xQ(x, y) e) ∀y∃xQ(x, y) f ) ∀x∀yQ(x, y)