AP Calculus BC 9.4 Defining and Differentiating Vector- Valued Functions Study Notes - New Syllabus

 Defining and Differentiating Vector-Valued Functions

A vector-valued function is a function whose output is a vector rather than a scalar. In two or three dimensions, the position of a moving object can be described as a vector function of time:

$ \mathbf{r}(t) = \langle x(t), \, y(t) \rangle \quad \text{(2D)},$

$\mathbf{r}(t) = \langle x(t), \, y(t), \, z(t) \rangle \quad \text{(3D)} $

Here, \( x(t) \), \( y(t) \), and \( z(t) \) are scalar functions of the parameter \( t \) (often time), representing the components of the vector along each axis.

Geometric Interpretation:

• The vector \( \mathbf{r}(t) \) represents the position of a particle at time \( t \).
• The derivative \( \mathbf{r}'(t) \) represents the velocity vector, indicating both speed and direction of motion.
• The second derivative \( \mathbf{r}”(t) \) represents the acceleration vector.

Definition of the Derivative:

The derivative of a vector-valued function is defined componentwise. If $ \mathbf{r}(t) = \langle x(t), \, y(t), \, z(t) \rangle $ then $ \mathbf{r}'(t) = \langle x'(t), \, y'(t), \, z'(t) \rangle $

The derivative measures the instantaneous rate of change of the position vector with respect to \( t \).

Magnitude of the Velocity (Speed):

The speed at time \( t \) is the magnitude of the velocity vector:

$ \text{Speed} = \left\| \mathbf{r}'(t) \right\| = \sqrt{ \left( x'(t) \right)^{2} + \left( y'(t) \right)^{2} + \left( z'(t) \right)^{2} } $

Key Notes:

• Differentiation rules for vector-valued functions follow the same rules as scalar calculus — derivatives are computed componentwise.
• Common rules include:
  • Sum Rule: \( \dfrac{d}{dt} \left[ \mathbf{u}(t) + \mathbf{v}(t) \right] = \mathbf{u}'(t) + \mathbf{v}'(t) \)
  • Scalar Multiple Rule: \( \dfrac{d}{dt} \left[ c \, \mathbf{u}(t) \right] = c \, \mathbf{u}'(t) \)
  • Product Rule (scalar × vector): \( \dfrac{d}{dt} \left[ f(t) \, \mathbf{u}(t) \right] = f'(t) \, \mathbf{u}(t) + f(t) \, \mathbf{u}'(t) \)
  • Dot Product Rule: \( \dfrac{d}{dt} \left[ \mathbf{u} \cdot \mathbf{v} \right] = \mathbf{u}’ \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{v}’ \)
  • Cross Product Rule: \( \dfrac{d}{dt} \left[ \mathbf{u} \times \mathbf{v} \right] = \mathbf{u}’ \times \mathbf{v} + \mathbf{u} \times \mathbf{v}’ \)
• The concepts of tangent lines, curvature, and motion analysis can be extended naturally to vector-valued functions.