Rates of Change in Parametric Functions
Introduced| English | Meaning |
|---|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ | The change in one quantity divided by the corresponding change in another. |
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ | A description of a curve or relationship in which coordinates or variables are expressed in terms of a parameter. |
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…
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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?
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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.
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Velocity is always tangent to the curve, pointing the way the object is currently moving.
Select all true statements about parametric rates of change.
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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 object has stopped moving completely.
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False. Only the vertical motion has momentarily stopped; $x$ may still be changing, as at the top of a thrown ball's arc.
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.
Write from a horizontal rate of 5
- x equals 5 t.
- The average horizontal rate is the coefficient.
A point has x equals 5 t. What is its average horizontal rate of change?
The change in x per unit t is 5.
What is the horizontal rate?
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A linear coefficient is the constant rate. Other options — 3 (the lesson rate); 5 t (the position formula).