Sketching Graphs of Functions and Their Derivatives · 绘制函数及其导数的图像
Draw the whole shape from its derivatives
- The derivatives are a recipe for a graph: $f'$ gives the slopes, $f''$ gives the bends.
- Combine the two sign charts and you can sketch a curve you were never handed.
- Extrema come from $f'$; concavity and inflection come from $f''$.
- This is where all of Unit 5 comes together into one picture.
从导数画出整个形状
- 导数是画图的配方:$f'$ 给斜率,$f''$ 给弯曲。
- 把两张符号表结合,你就能画出一条从未给你的曲线。
- 极值来自 $f'$;凹凸性与拐点来自 $f''$。
- 这里,整个第 5 单元汇成一幅图。
The information you assemble
- From $f'$: intervals of increase/decrease and the extrema (where $f'$ changes sign).
- From $f''$: intervals of concavity and the points of inflection (where $f''$ changes sign).
- Plus a few plotted points and any intercepts to anchor the height.
- Layer these together and the shape is forced.
你要汇总的信息
- 来自 $f'$:递增/递减区间和极值($f'$ 变号之处)。
- 来自 $f''$:凹凸区间和拐点($f''$ 变号之处)。
- 再加几个描点和任何截距来锚定高度。
- 把这些层层叠起,形状就被确定了。
A local extremum of $f$ occurs at a ____ of $f'$ where it changes sign. · $f$的局部极值发生在$f'$的一个____处，此处它改变了符号。
A sign-changing zero of $f'$ marks an extremum of $f$. · $f'$的变号零点标记了$f$的一个极值。
Which features come from $f''$ (the second derivative)? · 哪些特征来自$f''$（二阶导数）？
Concavity + inflection from $f''$; increase/decrease + extrema from $f'$. · 凹凸性和拐点来自$f''$；增减性和极值来自$f'$。
Sketching $f'$ from $f$, and back
- Given the graph of $f$: where $f$ has a horizontal tangent, $f'=0$ (a zero of $f'$).
- Where $f$ rises, $f'$ is positive; where $f$ falls, $f'$ is negative.
- Where $f$ is steepest, $f'$ is at an extreme; at an inflection of $f$, $f'$ has a max or min.
- Reverse the reading to sketch $f$ from a given $f'$.
从 $f$ 画 $f'$,以及反过来
- 给定 $f$ 的图:$f$ 有水平切线之处,$f'=0$($f'$ 的零点)。
- $f$ 上升之处,$f'$ 为正;$f$ 下降之处,$f'$ 为负。
- $f$ 最陡之处,$f'$ 取极值;在 $f$ 的拐点处,$f'$ 有最大或最小。
- 反过来读,就能从给定的 $f'$ 画出 $f$。
Where the graph of $f$ has a horizontal tangent, the graph of $f'$... · 当$f$的图像有水平切线时，$f'$的图像...
Horizontal tangent ⇒ slope $0$ ⇒ $f'=0$. · 水平切线 ⇒ 斜率$0$ ⇒ $f'=0$。
Where · 何地 $f$ is increasing, its derivative $f'$ is positive. · 当$f$递增时，其导数$f'$为正。
Rising ⇔ $f'>0$. · 上升 ⇔ $f'>0$。
Put the marks on the sketch
- Mark each extremum (peak/valley), each inflection point (bend flip), and label the intervals.
- Match increasing+concave-up (curving up steeply) vs. increasing+concave-down (leveling off), and so on — four combinations.
- Check consistency: a local max should sit where increase turns to decrease and the curve is concave down.
- A clean sketch shows all four features agreeing.
把标记画到草图上
- 标出每个极值(峰/谷)、每个拐点(弯曲翻转),并标注各区间。
- 区分递增+上凹(陡峭上弯)与递增+下凹(趋平)等——四种组合。
- 检查一致性:局部最大应位于递增转递减且曲线下凹之处。
- 一幅干净的草图会显示四个特征彼此吻合。
The finished shape · 最终形状
y = ax³ + cx
Peak at $(-1,2)$, inflection at $(0,0)$, valley at $(1,-2)$ — exactly what the $f'$ and $f''$ charts predict. · 峰值在$(-1,2)$，拐点在$(0,0)$，谷底在$(1,-2)$——这正是$f'$和$f''$图表所预测的。
At a point where $f$ is high but flat, the value of $f'$ is... · 在$f$高但平坦的点，$f'$的值是...
Read the slope, not the height: flat ⇒ $f'=0$. · 读取斜率而非高度：平坦 ⇒ $f'=0$。
A local maximum should occur where $f$ turns from increasing to decreasing and the curve is concave... · 局部最大值应出现在$f$从递增变为递减且曲线凹...
A peak is concave down (a cap). · 峰值是凹向下的（帽子状）。
Don't read $f'$ off the height of $f$ — read it off the slope. Where $f$ is high but flat, $f'=0$, not "large." And a zero of $f'$ marks a possible extremum of $f$; a zero of $f''$ marks a possible inflection of $f$ — keep the two levels straight.
别从 $f$ 的高度读 $f'$——要从斜率读。$f$ 高却平之处,$f'=0$,而非"很大"。而 $f'$ 的零点标记 $f$ 的一个可能极值;$f''$ 的零点标记 $f$ 的一个可能拐点——把这两个层次分清。
Sketch clues for $f(x)=x^3-3x$.
- $f'=3(x-1)(x+1)$: increasing on $(-\infty,-1)$ and $(1,\infty)$, decreasing on $(-1,1)$; max at $(-1,2)$, min at $(1,-2)$.
- $f''=6x$: concave down for $x<0$, concave up for $x>0$; inflection at $(0,0)$.
- The sketch: rises to a peak at $(-1,2)$, falls through $(0,0)$, dips to $(1,-2)$, then rises.
$f(x)=x^3-3x$ 的草图线索。
- $f'=3(x-1)(x+1)$:在 $(-\infty,-1)$ 与 $(1,\infty)$ 递增,在 $(-1,1)$ 递减;最大在 $(-1,2)$,最小在 $(1,-2)$。
- $f''=6x$:$x<0$ 下凹,$x>0$ 上凹;拐点在 $(0,0)$。
- 草图:升到 $(-1,2)$ 的峰,经 $(0,0)$ 下落,降到 $(1,-2)$ 的谷,然后再升。
To sketch a graph, combine what $f'$ says (increase/decrease and extrema) with what $f''$ says (concavity and inflection points), plus a few anchor points. Read $f'$ from the slope of $f$ (not its height); a zero of $f'$ is a possible extremum, a zero of $f''$ a possible inflection.
要画图,把 $f'$ 说的(递增/递减与极值)和 $f''$ 说的(凹凸性与拐点)结合,再加几个锚点。从 $f$ 的斜率(而非高度)读 $f'$;$f'$ 的零点是可能极值,$f''$ 的零点是可能拐点。