Rates of Change in Parametric Functions · 参数函数中的变化率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| rate of change/reɪt ɒv tʃeɪndʒ/ | 变化率 | biàn huà lǜ |
| parametric function/ˌpærəˈmetrɪk ˈfʌŋkʃn/ | 参数方程函数 | cān shù fāng chéng hán shù |
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.
多快,朝哪个方向?
- 知道球每一刻在哪里很好;知道它运动得多快更好。
- 速度大小和方向合起来构成速度矢量——两者都来自变化率。
- 对一条参数路径,我们问 $x$ 和 $y$ 各自随时间变化得多快。
- 这两个变化率组合成运动点的速度。
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 $\dfrac{\Delta y}{\Delta t}$.
两个坐标的变化率
- 水平变化率(rate of change)是 $\dfrac{dx}{dt}$——点横向运动多快。
- 竖直变化率是 $\dfrac{dy}{dt}$——它向上或向下运动多快。
- 对参数方程函数(parametric function),这两个就是速度的两个分量。
- 在一段区间上,平均变化率是 $\dfrac{\Delta x}{\Delta t}$ 或 $\dfrac{\Delta y}{\Delta t}$。
For a moving point, $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$ are the… · 对于一个移动的点，$\dfrac{dx}{dt}$和$\dfrac{dy}{dt}$分别是……
The rate of change of each coordinate is a velocity component: $dx/dt$ across, $dy/dt$ up. · 每个坐标的变化率是一个速度分量：$dx/dt$表示水平方向，$dy/dt$表示垂直方向。
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.
速度与路径相切
- 把两个分量组合起来,你就得到一个速度箭头。
- 那个箭头总是指向曲线的切线方向——当前的运动方向。
- 箭头长表示运动快;箭头短表示慢。
- 沿路径跟随这些箭头,就看出物体在加速和减速。
A point has $x = 3t$. What is its average horizontal rate of change $\Delta x/\Delta t$ over any interval? · 一个点具有$x = 3t$。在任意区间内，其平均水平变化率$\Delta x/\Delta t$是多少？
Since $x = 3t$ is linear, $\Delta x/\Delta t = 3$ everywhere — a constant horizontal velocity. · 由于$x = 3t$是线性的，$\Delta x/\Delta t = 3$处处——即恒定的水平速度。
The velocity vector points ____ to the path — along the direction of motion. · 速度矢量指向____于路径——沿运动方向。
Velocity is always tangent · 相切 to the curve, pointing the way the object is currently moving. · 速度始终与曲线相切，指向物体当前运动的方向。
Select all · 所有 true statements about parametric rates of change. · 选择关于参数变化率的所有正确陈述。
Rates measure change · 变化, not position. The other three are correct. · 变化率衡量的是变化而非位置。其他三项是正确的。
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.
当变化率为零时
- 如果某一瞬间 $\dfrac{dy}{dt} = 0$,竖直运动一时停止了。
- 那正是抛球弧线的最高点,它在下落前停止上升。
- 如果 $\dfrac{dx}{dt} = 0$,水平运动暂停——路径一时是竖直的。
- 这些零变化率的时刻,标志着运动的转折点。
Rate of change on a curve · 曲线上某点的变化率
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). · 切线的斜率即为变化率；对于参数曲线，它是(dy/dt)除以(dx/dt)。
If $dy/dt = 0$ at some instant, the vertical motion has momentarily stopped. · 如果在某一时刻$dy/dt = 0$，则垂直运动暂时停止。
A zero vertical rate means $y$ is momentarily not changing — the top of a thrown ball's arc, for example. · 零垂直变化率意味着$y$暂时不发生变化——例如投掷球弧线的顶点。
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.
解释这场运动
- $dx/dt$ 为正:向右运动;为负:向左。竖直方向同理。
- 比较两个变化率,看点是陡峭上升还是横向漂移。
- 恒定的变化率意味着直线、匀速运动;变化的变化率意味着弯曲或加速。
- 变化率把静态的路径变成对运动的完整描述。
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.
变化率 $dx/dt$ 是一个速度,而不是位置或距离。一个点可以离原点很远却运动得慢,或离原点很近却运动得快——位置和变化率是相互独立的。
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.
一个点有 $x = 3t$、$y = 20t - 5t^2$。
- 水平变化率:$\dfrac{dx}{dt} = 3$(恒定)——稳定的横向速度。
- 竖直变化率:$\dfrac{dy}{dt} = 20 - 10t$,在 $t = 2$ 时为 $0$。
- 所以在 $t = 2$ 时,球一时既不上升也不下降——它弧线的最高点。
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.
在参数方程函数中,变化率 $\dfrac{dx}{dt}$ 和 $\dfrac{dy}{dt}$ 是水平和竖直的速度分量。它们的组合与路径相切,而零变化率标志着运动方向一时停止之处。