Evaluate for .

Evaluate for r(t) = 〈sin t, 1 − cos t, −2t〉.

r(t) = 〈t, t², t³〉

r(t) = 〈e^3s, e^−s, s⁴〉

c(t) = t⁻¹i − e^2t k

a(θ) = (cos 3θ)i + (sin² θ)j + (tan θ)k

Calculate r′(t) and r″(t) for r(t) = 〈t, t², t³〉.

Stetch the curve r(t) = 〈1 − t², t〉 for −1 ≤ t ≤ 1. Compute the tangent vector at t = 1 and add it to the sketch.

Stetch the curve r₁(t) = 〈t, t²〉 together with its tangent vector at t = 1. Then do the same for r₂(t) = 〈t³, t⁶〉.

Sketch the cycloid r(t) = 〈t − sin t, 1 − cos t〉 together with its tangent vectors at and .

, assuming that

Compute in two ways:

Compute in two ways:

Let r(t) = 〈t², 1 − t, 4t〉. Calculate the derivative of r(t) · a(t) at t = 2, assuming that a(2) = 〈1, 3, 3〉 and a′(2) = 〈−1, 4, 1〉.

Let v(s) = s²i + 2sj + 9s⁻²k. Evaluate at s = 4, assuming that g(4) = 3 and g′(4) = −9.

Use Example 4 to calculate , where r(t) = 〈?t, t², e^t〉.

Let r(t) = 〈3 cos t, 5 sin t, 4 cos t〉. Show that ∥r(t)∥ is constant and conclude, using Example 7, that r(t) and r′(t) are orthogonal. Then compute r′(t) and verify directly that r′(t) is orthogonal to r(t).

Show that the derivative of the norm is not equal to the norm of the derivative by verifying that ∥r(t)∥′ ≠ ∥r(t)′∥ for r(t) = 〈t, 1, 1〉.

Show that for any constant vector a.

Find the location at t = 3 of a particle whose path (Figure 8) satisfies

Find the location and velocity at t = 4 of a particle whose path satisfies

A fighter plane, which can shoot a laser beam straight ahead, travels along the path r(t) = 〈5 − t, 21 − t², 3 − t³/27〉. Show that there is precisely one time t at which the pilot can hit a target located at the origin.

The fighter plane of Exercise 57 travels along the path r(t) = 〈t − t³, 12 − t², 3 − t〉. Show that the pilot cannot hit any target on x-axis.

Find all solutions to r′(t) = v with initial condition r(1) = w, where v and w are constant vectors in R³.

Let u be a constant vector in R³. Find the solution of the equation r′(t) = (sin t)u satisfying r′(0) = 0.

Find all solutions to r′(t) = 2r(t) where r(t) is a vector-valued function in three-space.

Show that w(t) = 〈sin(3t + 4), sin(3t − 2), cos 3t〉 satisfies the differential equation w″(t) = −9w(t).

Prove that the Bernoulli spiral (Figure 9) with parametrization r(t) = 〈e^t cos 4t, e^t sin 4t〉 has the property that the angle ψ between the position vector and the tangent vector is constant. Find the angle ψ in degrees.

A curve in polar form r = f(θ) has parametrization

r(θ) = f(θ) 〈cos θ, sin θ〉

Let ψ be the angle between the radial and tangent vectors (Figure 10). Prove that

Hint: Compute r(θ) × r′(θ) and r(θ) · r′(θ).

Prove that if ∥r(t)∥ takes on a local minimum or maximum value at t₀, then r(t₀) is orthogonal to r′(t₀). Explain how this result is related to Figure 11. Hint: Observe that if ∥r(t₀)∥ is a minimum, then r(t) is tangent at t₀ to the sphere of radius ∥r(t₀)∥ centered at the origin.

Newton’s Second Law of Motion in vector form states that where F is the force acting on an object of mass m and p = mr′(t) is the object’s momentum. The analogs of force and momentum for rotational motion are the torque τ = r × F and angular momentum

J = r(t) × p(t)

Use the Second Law to prove that .