Vector-Valued Functions Free Online FlashCards

What does a vector-valued function assign?

A vector-valued function assigns a vector to each real parameter value, usually written r(t) = ⟨x(t), y(t), z(t)⟩.

How is a vector-valued limit computed?

Take the limit of every component separately: limₜ→ₐ r(t) = ⟨limₜ→ₐ f(t), limₜ→ₐ g(t), limₜ→ₐ h(t)⟩.

When is a vector-valued function continuous?

A vector-valued function is continuous at t = a when limₜ→ₐ r(t) = r(a), equivalently when every component is continuous there.

How do you differentiate a vector-valued function?

Differentiate each component: if r(t) = ⟨f(t), g(t), h(t)⟩, then r′(t) = ⟨f′(t), g′(t), h′(t)⟩.

How are velocity and acceleration related to position?

If r(t) is position, then velocity is v(t) = r′(t), and acceleration is a(t) = v′(t) = r″(t).

What is the speed of a particle?

Speed is the magnitude of velocity: ‖v(t)‖ = ‖r′(t)‖. It is a nonnegative scalar, unlike velocity, which is a vector.

Does constant speed imply zero acceleration?

No. A particle can have constant speed but nonzero acceleration when its velocity changes direction, as in uniform circular motion.

What does r′(t₀) represent geometrically?

If r′(t₀) ≠ 0, then r′(t₀) is a tangent vector at the point r(t₀). It gives the curve’s instantaneous direction.

What is the vector equation of a tangent line?

The tangent line is L(s) = r(t₀) + s r′(t₀), where s is a new real parameter.

How is the unit tangent vector found?

The unit tangent vector is T(t) = r′(t) / ‖r′(t)‖, provided r′(t) ≠ 0.

How is dy/dx found from a parametric curve?

When x′(t) ≠ 0, dy/dx = (dy/dt)/(dx/dt) = y′(t)/x′(t).

How can position be recovered from velocity?

Recover position using r(t) = r(t₀) + ∫ₜ₀ᵗ v(u) du, integrating each vector component separately.