Vector-Valued Functions in AP Calculus BC: From Curves to Motion

Vector-valued function

A function whose output is a vector (e.g., ⟨x(t), y(t)⟩ or ⟨x(t), y(t), z(t)⟩), often used to represent a particle’s position in 2D/3D over time.

Parameter (t)

An input variable (often time) that drives the component functions in a parametric/vector description of a curve.

Component functions

The scalar functions x(t), y(t), and z(t) that make up a vector-valued function r(t)=⟨x(t), y(t), z(t)⟩.

Position vector

The vector r(t) from the origin to the point (x(t), y(t)) or (x(t), y(t), z(t)) at time/parameter t.

Parametric equations

A way to describe a curve by giving coordinates as functions of a parameter: x=x(t), y=y(t) (and possibly z=z(t)).

Vector form of a parametric curve

Writing parametric equations as a single vector function, e.g., r(t)=\langle x(t), y(t) angle, representing the same geometric path.

Vertical line test (limitation)

A test for whether a curve is a function $y=f(x)$; many paths (like circles) fail it but can still be represented parametrically.

Componentwise limit (vector-valued)

A vector limit computed by taking limits of each component: $\lim_{t\to a} r(t)=\langle \lim x(t), \lim y(t) \rangle$ (and similarly in 3D).

Continuity of a vector-valued function

r(t) is continuous at t=a if each component function is continuous at t=a; any component discontinuity makes r(t) discontinuous.

Differentiating vector-valued functions (componentwise rule)

To differentiate r(t)=\langle x(t), y(t), z(t) angle, differentiate each component: r′(t)=\langle x′(t), y′(t), z′(t) angle.

Derivative notation for vector-valued functions

Common ways to write the derivative include r′(t) and d r/dt, meaning the rate of change of the vector with respect to t.

Tangent (direction) vector

For a position function r(t), the derivative r′(t) points in the instantaneous direction of motion along the curve and is tangent to the path.

Parametric slope formula (dy/dx)

For x=x(t), y=y(t), the slope in the xy-plane is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided $\frac{dx}{dt} ≠ 0$.

Velocity vector

In motion problems, v(t)=r′(t); it gives both the instantaneous speed and direction of motion.

Acceleration vector

a(t)=v′(t)=r″(t); it describes how velocity changes over time (in magnitude and/or direction).

Speed

A scalar equal to the magnitude of velocity: |v(t)| (not the velocity vector itself).

Magnitude of velocity in 2D

If v(t)=\langle x′(t), y′(t) angle, then |v(t)| = \sqrt{(x′(t))^2 + (y′(t))^2}.

Magnitude of velocity in 3D

If v(t)=\langle x′(t), y′(t), z′(t) angle, then |v(t)| = \sqrt{(x′(t))^2 + (y′(t))^2 + (z′(t))^2}.

Displacement

The change in position from t=a to t=b: r(b) − r(a) (a vector).

Distance traveled (total path length)

The total length of the path from t=a to t=b: $\int_a^b |v(t)| \, dt$ (a scalar).

Recovering position from velocity

If $v(t) = r'(t)$, then $r(t)$ can be found by integrating componentwise: $r(t) = \int v(t) dt$, then using an initial condition to find constants.

Initial condition (position)

A given value like r(t0)=⟨x0, y0⟩ used to determine constants of integration when finding r(t) from v(t).

Constants of integration (vector case)

When integrating a vector-valued function, each component may require its own constant (e.g., C_1, C_2, C_3).

Common derivative misconception: |r(t)| vs |v(t)|

|r(t)| is distance from the origin, not speed; speed is |v(t)|, the magnitude of the velocity vector.

Common derivative misconception: d/dt(|r(t)|) vs |r′(t)|

In general, d/dt(|r(t)|) ≠ |r′(t)| because the left is a derivative of a scalar magnitude, while the right is the magnitude of a derivative vector.