Vector-Valued Functions

Unit 9 · Parametric, Polar & Vector

Vector-Valued Functions

A vector-valued function r(t) = ⟨x(t), y(t)⟩ has derivative r′(t) = ⟨x′(t), y′(t)⟩ (velocity) and integral ∫r(t) dt = ⟨∫x dt, ∫y dt⟩. Differentiation and integration are done component by component, with the same rules as scalar calculus.

Derivative and Integral

\vec{r}'(t) = \langle x'(t),\; y'(t) \rangle, \qquad \vec{r}''(t) = \langle x''(t),\; y''(t) \rangle

\int \vec{r}(t)\,dt = \left\langle \int x(t)\,dt,\; \int y(t)\,dt \right\rangle + \vec{C}

Position from Velocity

\vec{r}(t) = \vec{r}(t_0) + \int_{t_0}^{t} \vec{v}(\tau)\,d\tau

AP Tip: Differentiate and integrate each component independently — treat angle-bracket notation like two parallel scalar problems.

Type 1

Differentiating Vector-Valued Functions

Differentiate each component separately to find velocity r′(t) and acceleration r″(t).

Example 1

Find r′(t) and r″(t) for r(t) = ⟨t² + 1, e^{3t}⟩.

Example 2

Find r′(t) and r″(t) for r(t) = ⟨3t² − t, ln(t+1)⟩.

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Type 2

Integrating Vector-Valued Functions

Integrate each component separately. For definite integrals, evaluate each component integral independently.

Example 3

Evaluate ∫₀² ⟨2t, eᵗ⟩ dt.

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Type 3

Finding Position from Velocity

Integrate v(t) component-by-component, then apply the initial condition r(t₀) = r₀ to find each constant.

Example 4

v(t) = ⟨cos t, 2t⟩ and r(0) = ⟨0, 1⟩. Find r(π).

Example 5

v(t) = ⟨3t², 2cos t⟩ and r(0) = ⟨−1, 4⟩. Find r(t).

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