Change in Tandem
Introduced| English |
|---|
| input/ˈɪnpʊt/ |
| output/ˈaʊtpʊt/ |
| increasing/ɪnˈkriːsɪŋ/ |
| decreasing/ˈdiːkriːsɪŋ/ |
| constant/ˈkɒnstənt/ |
| interval/ˈɪntəvl/ |
| concavity/kənˈkævɪti/ |
| rate of change/reɪt ɒv tʃeɪndʒ/ |
| concave up/kɒnˈkeɪv ʌp/ |
| concave down/kɒnˈkeɪv daʊn/ |
| turning point/ˈtɜːnɪŋ pɔɪnt/ |
| local maximum/ˈləʊkl ˈmæksɪməm/ |
| local minimum/ˈləʊkl ˈmɪnɪməm/ |
Watch the water climb
- Pour water into a glass at a steady rate and watch the level rise.
- In a straight glass the level climbs steadily; in a curvy vase it speeds up and slows down.
- Two quantities — the water poured and the height — change together, in tandem.
- Precalculus starts by describing exactly how one quantity changes as another one does.
A function links an input to an output
- A function takes an input 输入 (a value of $x$) and returns exactly one output 输出 (written $f(x)$).
- As the input changes, the output changes with it — that is change in tandem.
- We write $f(2)$ to mean "the output when the input is $2$".
- So the whole story of a function is one question: as $x$ moves, where does $f(x)$ go?
Increasing, decreasing, or constant
- Pick an interval 区间 — a stretch of the input axis — and follow the output from left to right.
- The function is increasing 递增 where the output rises as the input moves right.
- It is decreasing 递减 where the output falls as the input moves right.
- It is constant 常数 where the output stays level.
- You can read this straight off a graph: uphill is increasing, downhill is decreasing.
A function is increasing on an interval when…
Increasing is about direction, not size: move right along the input and the output climbs. It can be increasing while its values are still negative.
If a graph is perfectly flat (horizontal) over an interval, the function is constant there.
A flat graph means the output does not change as the input changes — that is exactly what constant means.
Concavity: is the change speeding up or slowing down?
- Even while a curve keeps increasing, its rate of change 变化率 (how fast the output moves) can itself change.
- Concavity 凹凸性 describes which way the curve bends.
- Concave up 凹 — the curve bends upward like a cup, so the rate of change is increasing.
- Concave down 凸 — the curve bends downward like a dome, so the rate of change is decreasing.
- The curvy vase again: where it narrows, the height climbs faster and faster (concave up).
Saying a curve is concave up tells you that…
Concave up bends like a cup: the rate of change itself is rising. A curve can be concave up while it is going down — think of a slide flattening out at the bottom.
Turning points: local maxima and minima
- Where a curve stops increasing and starts decreasing, it reaches a local maximum 极大值 — a peak.
- Where it stops decreasing and starts increasing, it reaches a local minimum 极小值 — a valley.
- Each such peak or valley is a turning point 转折点.
- "Local" means highest or lowest nearby — not necessarily over the whole graph.
The tangent's slope IS the rate of change
f(x) = x³ − 3x
Drag the point and read the tangent's slope — positive where the curve rises, negative where it falls, and zero at a turning point.
A point where a graph stops increasing and starts decreasing is a local ____.
At a local maximum the curve turns from uphill to downhill — a peak that is highest compared with nearby points.
Look at the curve in the figure. Select all the true statements.
The curve rises, turns at a peak, falls to a valley, then rises again. It is not concave up everywhere — it bends down near the peak and up near the valley.
"Increasing" and "concave up" are different ideas. Increasing is about the output going up; concave up is about the rate of change going up. A curve can be decreasing and concave up at the same time — falling, but flattening out as it falls.
Comparing two outputs
- To compare a function at two inputs $a$ and $b$, compare their outputs $f(a)$ and $f(b)$.
- If $b > a$ and $f(b) > f(a)$, the output grew as the input grew.
- The net change in output from $a$ to $b$ is simply $f(b) - f(a)$.
- If the function returns to the same value, the net change is zero — even if it rose and fell along the way.
A function has $f(2) = 5$ and $f(6) = 5$. What is the net change in output from $x = 2$ to $x = 6$?
Net change $= f(6) - f(2) = 5 - 5 = 0$. The output ended where it started — even though it may have risen and fallen in between.
Match each term to what it describes.
Increasing/decreasing describe the output's direction; concavity describes how the rate of change itself is changing; a local maximum is where the direction flips.
A cup of tea cools from 90°C to room temperature.
- Temperature (output) is decreasing the whole time — it never goes back up.
- But it falls fast at first, then slowly: the curve is concave up.
- There is no local maximum or minimum in between — the direction never flips.
A function's behaviour is read from its graph: it is increasing or decreasing on intervals, its concavity tells you whether the rate of change is speeding up or slowing down, and its turning points are the local maxima and minima where the direction flips.