AP® Calculus BC · Unit 9 · Topic 9.4
Defining and Differentiating Vector-Valued Functions
A vector-valued function packages x(t) and y(t) into one position arrow r(t); differentiating each component gives the velocity vector, and differentiating again gives acceleration.
A vector-valued function r(t) = (x(t), y(t)) treats a moving point as an arrow from the origin, growing or swinging as t changes. This beats tracking x(t) and y(t) as separate stories, because motion is really one thing happening in two directions at once. The arrow's tip traces the path, and the arrow itself carries direction and distance from the origin at every instant.
Differentiate component by component: r'(t) = (x'(t), y'(t)) is the velocity vector, tangent to the path and pointing the way the object is currently heading. Its magnitude |r'(t)| = √(x'(t)2 + y'(t)2) is speed, a plain number with no direction. Differentiate again to get r''(t) = (x''(t), y''(t)), the acceleration vector, which tracks how the velocity arrow itself is changing — in length, direction, or both. None of this is new calculus; it's the same derivative rules applied twice, separately, to two coordinate functions riding together.
This connects straight back to tangent lines on parametric curves: dy/dx = y'(t)/x'(t) is just the slope direction of the velocity vector r'(t), stripped of magnitude. A cusp shows up exactly when x'(t) and y'(t) are both zero at once — the velocity arrow shrinks to nothing and the path can turn on a dime. Vector functions also set up the next idea cleanly: once you have r'(t) as a vector, arc length and curvature are just measurements built from its magnitude and how fast its direction turns.
Key terms
- position vector
- The vector r(t) = (x(t), y(t)) pointing from the origin to the location of a moving object at time t.
- velocity vector
- r'(t), the component-wise derivative of position, tangent to the path and pointing in the direction of motion.
- speed
- The magnitude |r'(t)| of the velocity vector, a single nonnegative number with no direction.
- acceleration vector
- r''(t), the component-wise second derivative of position, describing how velocity changes.
- cusp
- A sharp point on a curve where x'(t) and y'(t) are simultaneously zero, so the velocity vector momentarily vanishes.
Watch it explained
- How it worksHow does differentiating a position vector component by component give a velocity arrow?
- Worked exampleFind the velocity and acceleration vectors for r(t) = (t2, 4t - t2) at t = 1.
- Common mistakeWhy isn't the derivative of a vector's length the same as the length of its derivative?
- Ties togetherHow does a sharp turn in the velocity arrow connect to the cusp condition found in the tangent line topic two sections back?
Check yourself
End of topic.