Parametric Functions and Rates of Change

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

Parametric AROC has two flavors. First, the AROC of each component with respect to $t$ : $(x(t_2) - x(t_1)) / (t_2 - t_1)$ and similarly for $y$ . Second, the AROC of $y$ with respect to $x$ — the slope of the secant line on the trajectory: $\Delta y / \Delta x$ . The sign tells you the direction of motion through the interval.

AROC formulas

AROC of x w.r.t. t

\frac{\Delta x}{\Delta t} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}

AROC of y w.r.t. x (secant slope)

\frac{\Delta y}{\Delta x} = \frac{y(t_2) - y(t_1)}{x(t_2) - x(t_1)}

AP Tip: The secant slope

\Delta y / \Delta x

is the AROC of

y

as a function of

x

along the trajectory — exactly what AROC of a Cartesian function would give.

Type 1

AROC of x or y with respect to t

Evaluate the component at both endpoints, subtract, divide by Δt.

Example 1

For

x(t) = t^2

, find AROC of

x

over

[1, 3]

. (A)

2

(B)

3

(C)

4

(D)

5

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Type 2

AROC of y with respect to x

Compute Δy and Δx separately, then divide.

Example 2

For

x = t, y = t^2

, find AROC of

y

with respect to

x

over

t \in [1, 3]

. (A)

2

(B)

3

(C)

4

(D)

5

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Type 3

Interpret sign of AROC

Positive Δx means moving right; positive Δy means moving up. Combine for direction.

Example 3

Between

t = 0

and

t = π/2

on the path

\vec r(t) = (\cos t, \sin t)

, in which general direction does the object move? (A) Up and right (B) Up and left (C) Down and right (D) Down and left

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