Unit 9 Review: Vector-Valued Functions in AP Calculus BC

Vector-valued function

A function whose domain is a set of real numbers and whose range is a set of vectors.

Component Form

A representation of vector-valued functions as ( \mathbf{r}(t) = \langle x(t), y(t) \rangle ).

Unit Vector Form

A representation of vector-valued functions as ( \mathbf{r}(t) = x(t)\mathbf{i} + y(t)\mathbf{j} ).

Position vector

The vector ( \mathbf{r}(t) ) describes the position of a point in space in terms of its coordinates at time ( t ).

Continuous function

A function is continuous at ( t=a ) if limits exist for both components and equal the function value at that point.

Derivative of vector-valued function

The derivative is computed component-wise as ( \mathbf{r}'(t) = \langle x'(t), y'(t) \rangle ).

Tangent vector

The derivative vector ( \mathbf{r}'(t) ) is tangent to the curve at the point corresponding to ( t ).

Indefinite integral of vector-valued function

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

Definite integral of vector-valued function

( \int{a}^{b} \mathbf{r}(t) \, dt = \left\langle \int{a}^{b} x(t) \, dt, \int_{a}^{b} y(t) \, dt \right\rangle ).

Velocity vector

Given by ( \mathbf{v}(t) = \mathbf{r}'(t) = \langle x'(t), y'(t) \rangle ).

Acceleration vector

Given by ( \mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t) = \langle x''(t), y''(t) \rangle ).

Speed

The magnitude of the velocity vector, computed as ( Speed = |\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2} ).

Displacement

A vector representing the net change in position, ( \int_{a}^{b} \mathbf{v}(t) \, dt = \langle x(b) - x(a), y(b) - y(a) \rangle ).

Total distance traveled

The arc length of the path traveled, calculated as ( Total \, Distance = \int_{a}^{b} |\mathbf{v}(t)| \, dt ).

Fundamental Theorem of Calculus for Vectors

To find position at ( t=b ): ( \mathbf{r}(b) = \mathbf{r}(a) + \int_{a}^{b} \mathbf{v}(t) \, dt ).

Confusion between velocity and speed

Velocity is a vector (has direction); speed is a scalar (magnitude of velocity).

Initial Conditions in Motion

Must include the starting position when calculating position at ( t=b ): ( \mathbf{r}(b) = \mathbf{r}(a) + \int_a^b \mathbf{v}(t) \, dt ).

Component-wise operations

Calculus operations on vector-valued functions are performed separately on ( x(t) ) and ( y(t) ).

Common notation error

Speed should not be written as a vector; speed is a scalar quantity.

Vector constant of integration

When integrating, the constant ( \mathbf{C} ) is a vector ( \langle C1, C2 \rangle ), not a scalar.

Particle motion application

Vector functions are commonly used to model particle motion in the plane, treating coordinates ( (x, y) ) as position over time.

Continuous at point

A vector function is continuous at point ( t=a ) if both components' limits exist and equal the value of the function at that point.

Arc length formula for distance

Total distance traveled can be computed as ( \int_{a}^{b} \sqrt{(x'(t))^2 + (y'(t))^2} \, dt ).

Position vector notation

Position vectors are often written in terms of their components: ( \mathbf{r}(t) = \langle x(t), y(t) \rangle ).

Differentiation of component functions

If ( x(t) ) and ( y(t) ) are differentiable, then ( \mathbf{r}'(t) = \langle x'(t), y'(t) \rangle ).