Rates of Change in Parametric Functions
Introduced| English |
|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ |
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ |
How fast, and which way?
- Knowing where a ball is at each moment is good; knowing how fast it moves is better.
- Speed and direction together make velocity — and both come from rates of change.
- With a parametric path, we ask how quickly $x$ and $y$ each change with time.
- Those two rates combine into the velocity of the moving point.
Rates of the two coordinates
- The horizontal rate of change 变化率 is $\dfrac{dx}{dt}$ — how fast the point moves sideways.
- The vertical rate is $\dfrac{dy}{dt}$ — how fast it moves up or down.
- For a parametric function 参数方程函数, these are the two velocity components.
- Over an interval, the average rate is $\dfrac{\Delta x}{\Delta t}$ or $\dfrac{\Delta y}{\Delta t}$.
For a moving point, $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ are the…
The rate of change of each coordinate is a velocity component: $dx/dt$ across, $dy/dt$ up.
Velocity is tangent to the path
- Combine the two components and you get a velocity arrow.
- That arrow always points tangent to the curve — the current direction of travel.
- A long arrow means fast motion; a short one means slow.
- Following the arrows along the path shows the object speeding up and slowing down.
A point has $x = 3t$. What is its average horizontal rate of change $\Delta x/\Delta t$ over any interval?
Since $x = 3t$ is linear, $\Delta x/\Delta t = 3$ everywhere — a constant horizontal velocity.
The velocity vector points ____ to the path — along the direction of motion.
Velocity is always tangent to the curve, pointing the way the object is currently moving.
Select all true statements about parametric rates of change.
Rates measure change, not position. The other three are correct.
When a rate is zero
- If $\dfrac{dy}{dt} = 0$ for an instant, the vertical motion has momentarily stopped.
- That is exactly the top of a thrown ball's arc, where it stops rising before falling.
- If $\dfrac{dx}{dt} = 0$, the horizontal motion pauses — the path is momentarily vertical.
- These zero-rate moments mark the turning points of the motion.
Rate of change on a curve
y = x^2
The slope of the tangent line is the rate of change; for a parametric curve it is (dy/dt) over (dx/dt).
If $dy/dt = 0$ at some instant, the vertical motion has momentarily stopped.
A zero vertical rate means $y$ is momentarily not changing — the top of a thrown ball's arc, for example.
Interpreting the motion
- Positive $dx/dt$: moving right; negative: moving left. Same idea vertically.
- Compare the two rates to see whether the point climbs steeply or drifts sideways.
- Constant rates mean straight-line, steady motion; changing rates mean curving or accelerating.
- The rates turn the static path into a full description of the movement.
The rate $dx/dt$ is a velocity, not a position or a distance. A point can be far from the origin yet moving slowly, or near it yet moving fast — position and rate of change are independent.
A point has $x = 3t$ and $y = 20t - 5t^2$.
- Horizontal rate: $\dfrac{dx}{dt} = 3$ (constant) — steady sideways speed.
- Vertical rate: $\dfrac{dy}{dt} = 20 - 10t$, which is $0$ at $t = 2$.
- So at $t = 2$ the ball is momentarily neither rising nor falling — the top of its arc.
In a parametric function, the rates of change $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ are the horizontal and vertical velocity components. Their combination is tangent to the path, and a zero rate marks where that direction of motion momentarily stops.