Determine the domains of the vector-valued functions.

Sketch the paths r₁(θ) = 〈θ, cos θ〉 and r₂(θ) = 〈cos θ, θ〉 in the xy-plane.

Find a vector parametrization of the intersection of the surfaces x² + y⁴ + 2z³ = 6 and x = y² in R³.

Find a vector parametrization using trigonometric functions of the intersection of the plane x + y + z = 1 and the elliptical cylinder in R³.

Calculate

Calculate

A particle located at (1, 1, 0) at time t = 0 follows a path whose velocity vector is v(t) = 〈1, t, 2t²〉. Find the particle’s location at t = 2.

Find the vector-valued function r(t) = 〈x(t), y(t)〉 in R² satisfying r′(t) = −r(t) with initial conditions r(0) = 〈1, 2〉.

Calculate r(t) assuming that

Solve r″(t) = 〈t² − 1, t + 1, t³〉 subject to the initial conditions r(0) = 〈1, 0, 0〉 and r′(0) = 〈−1, 1, 0〉

Compute the length of the path

Express the length of the path r(t) = 〈ln t, t, e^t 〉 for 1 ≤ t ≤ 2 as a definite integral, and use a computer algebra system to find its value to two decimal places.

Find an arc length parametrization of a helix of height 20 cm that makes four full rotations over a circle of radius 5 cm.

Find the minimum speed of a particle with trajectory r(t) = 〈t, e^t−3,e^4−t〉.

A projectile fired at an angle of 60° lands 400 m away. What was its initial speed?

A specially trained mouse runs counterclockwise in a circle of radius 0.6 m on the floor of an elevator with speed 0.3 m/s while the elevator ascends from ground level (along the z-axis) at a speed of 12 m/s. Find the mouse’s acceleration vector as a function of time. Assume that the circle is centered at the origin of the xy-plane and the mouse is at (2, 0, 0) at t = 0.

During a short time interval [0.5, 1.5], the path of an unmanned spy plane is described by

A laser is fired (in the tangential direction) toward the yz-plane at time t = 1. Which point in the yz-plane does the laser beam hit?

A force F = 〈12t + 4, 8 − 24t〉 (in newtons) acts on a 2-kg mass. Find the position of the mass at t = 2 s if it is located at (4, 6) at t = 0 and has initial velocity 〈2, 3〉 in m/s.

Find the unit tangent vector to r(t) = 〈sin t, t, cos t〉 at t = π.

Find the unit tangent vector to r(t) = 〈t², tan⁻¹ t, t〉 at t = 1.

Calculate κ(1) for r(t) = 〈ln t, t〉.

Calculate for r(t) = 〈tan t, sec t, cos t〉.

At a certain time t₀, the path of a moving particle is tangent to the y-axis. The particle’s speed at time t₀ is 4 m/s, and its acceleration vector is a = 〈5, 4, 12〉. Determine the curvature of the path at t₀.

Parametrize the osculating circle to y = x² − x³ at x = 1.

Parametrize the osculating circle to at x = 4.

If a planet has zero mass (m = 0), then Newton’s laws of motion reduce to r″(t) = 0 and the orbit is a straight line r(t) = r₀ + tv₀, where r₀ = r(0) and v₀ = r′(0) (Figure 1). Show that the area swept out by the radial vector at time t is and thus Kepler’s Second Law continues to hold (the rate is constant).

Suppose the orbit of a planet is an ellipse of eccentricity e = c/a and period T (Figure 2). Use Kepler’s Second Law to show that the time required to travel from A′ to B′ is equal to

The period of Mercury is approximately 88 days, and its orbit has eccentricity 0.205. How much longer does it take Mercury to travel from A′ to B′ than from B′ to A (Figure 2)?