AP Calculus BC / Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
9.4 Defining and Differentiating Vector-Valued Functions
Notes
A vector function $\vec r(t)=\langle x(t),y(t)\rangle$ gives position. Differentiate each component:
- Velocity $\vec v=\langle x',y'\rangle$; acceleration $\vec a=\langle x'',y''\rangle$.
Practice
$\vec r(t)=\langle t^2, \sin t\rangle$. $\vec v(t)=$
Componentwise derivatives.
For that same $\vec r$, $\vec a(t)=$
Differentiate again.
$\vec r(t)=\langle e^t, t^3\rangle$. $\vec v(0)=$
$\langle e^0, 3(0)^2\rangle$.
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