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9.4 Defining and Differentiating Vector-Valued Functions

Part of Parametric Equations, Polar Coordinates, and Vector-Valued Functions in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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A vector-valued function gives position in the plane or space using component functions. In two dimensions, it often looks like
r(t)=⟨x(t),y(t)⟩\mathbf{r}(t)=\langle x(t),y(t)\rangle
and in three dimensions,
r(t)=⟨x(t),y(t),z(t)⟩\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle
You differentiate a vector-valued function component by component, so
r′(t)=⟨x′(t),y′(t)⟩\mathbf{r}'(t)=\langle x'(t),y'(t)\rangle
or similarly in three dimensions. This derivative is often interpreted as velocity.

In this unit

9 lessons

  1. 9.1Defining and Differentiating Parametric EquationsPreview lesson
  2. 9.2Second Derivatives of Parametric EquationsPreview lesson
  3. 9.3Finding Arc Lengths of Curves Given by Parametric EquationsPreview lesson
  4. 9.4Defining and Differentiating Vector-Valued FunctionsThis lesson
  5. 9.5Integrating Vector-Valued FunctionsPreview lesson
  6. 9.6Solving Motion Problems Using Parametric and Vector-Valued FunctionsPreview lesson
  7. 9.7Defining Polar Coordinates and Differentiating in Polar FormPreview lesson
  8. 9.8Find the Area of a Polar Region or the Area Bounded by a Single Polar CurvePreview lesson
  9. 9.9Finding the Area of the Region Bounded by Two Polar CurvesPreview lesson

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