虽然切线是一个非常有用的工具,但是在调查函数的图表方面,切线线无法说明图表如何在一个点处“弯曲”。这是第二阶段发挥的地方。据/p>
一种据strong>二阶衍生据/strong>是如何如何改变数量的变化率如何变化,这是通过再次分化量的第一衍生来获得的。据/p>
符号:据/strong>
- 函数的第二个衍生物据span class="katex">
F据/span>(据/span>X据/span>)据/span>是写作的据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>。据/li>
- 在莱布尼兹的符号中,第二个衍生物据span class="katex">
y据/span>(依赖变量)相对于据span class="katex">
X据/span>(独立变量)被写为据span class="katex">
D.据/span>X据/span>2据/span>D.据/span>2据/span>y据/span>=据/span>D.据/span>X据/span>D.据/span>(据/span>D.据/span>X据/span>D.据/span>y据/span>)据/span>。据/span>
找到该功能的第二个衍生品据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>一种据/span>X据/span>, 在哪里据span class="katex">
一种据/span>是任何常数。据/p>
我们有据/p>
F据/span>(据/span>X据/span>)据/span>=据/span>一种据/span>X据/span>那据/span>F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>一种据/span>那据/span>F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>。据/span>
因此,答案是0。据span class="katex">
□据/span>
找到该功能的第二个衍生品据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>1据/span>。据/p>
我们有据/p>
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>1据/span>那据/span>F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>X据/span>2据/span>-据/span>1据/span>那据/span>F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>X据/span>3.据/span>2据/span>。据/span>
因此,答案是据span class="katex">
X据/span>3.据/span>2据/span>。据span class="katex">
□据/span>
与图形的关系:据/strong>
首先,我们将列出与图形和第二衍生物相关的所有定容和点,然后移动到示例(从基本到高级)。据/p>
凹凸定理:据/p>
如果是功能据span class="katex">
F据/span>是两次可分辨率据span class="katex">
X据/span>=据/span>一种据/span>,然后是图表据span class="katex">
F据/span>凹陷在据span class="katex">
(据/span>一种据/span>那据/span>F据/span>(据/span>一种据/span>)据/span>)据/span>如果据span class="katex">
F据/span>'据/span>'据/span>(据/span>一种据/span>)据/span>>据/span>0.据/span>并且凹陷如果据span class="katex">
F据/span>'据/span>'据/span>(据/span>一种据/span>)据/span>据据/span>0.据/span>。据/p>
拐点:据/p>
拐点是曲线的曲线上的点的点,曲率变化的凹陷(符号)。拐点可能是静止的,但它们不是局部最大值或最小值。据/p>
要找到一个拐点,让我们看看一个例子:据/p>
(基本的)据/p>
让据span class="katex">
F据/span>是一个定义的函数据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>3.据/span>-据/span>3.据/span>X据/span>2据/span>+据/span>2据/span>X据/span>+据/span>1据/span>。找到图表的不同部分的凹面据span class="katex">
F据/span>(据/span>X据/span>)据/span>对于不同的值据span class="katex">
X据/span>。还要计算拐点。据/p>
我们有据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>6.据/span>X据/span>-据/span>6.据/span>。据/p>
为了据span class="katex">
X据/span>>据/span>1据/span>那据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>。因此,曲线是凹陷的据span class="katex">
X据/span>>据/span>1据/span>。据B.r>为了据span class="katex">
X据/span>据据/span>1据/span>那据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>据据/span>0.据/span>。因此,图形是凹陷的据span class="katex">
X据/span>据据/span>1据/span>。据B.r>由于凹部对面据span class="katex">
X据/span>>据/span>1据/span>和据span class="katex">
X据/span>据据/span>1据/span>,凹部在哪里变化据span class="katex">
X据/span>=据/span>1据/span>。据span class="katex">
□据/span>
笔记:据/strong>我们可以推断出拐点可以获得据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>,这是拐点的必要条件。据/p>
(中间的)据/p>
绘制图表据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>晒黑据/span>X据/span>为了据span class="katex">
X据/span>∈据/span>(据/span>2据/span>-据/span>π据/span>那据/span>2据/span>π据/span>)据/span>。据/p>
我们有据/p>
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>秒据/span>2据/span>X据/span>那据/span>F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>2据/span>晒黑据/span>X据/span>秒据/span>2据/span>X据/span>。据/span>
- 拐点:据B.r>由于拐点通过设置获得据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>, 我们有据span class="katex">
X据/span>=据/span>0.据/span>因为据span class="katex">
X据/span>∈据/span>(据/span>2据/span>-据/span>π据/span>那据/span>2据/span>π据/span>)据/span>。据/span>因此拐点是据span class="katex">
(据/span>0.据/span>那据/span>0.据/span>)据/span>。据/li>
- 凹:据B.r>自从据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>=据/span>2据/span>晒黑据/span>X据/span>秒据/span>2据/span>X据/span>,我们有以下几点:据你l>
- 为了据span class="katex">
X据/span>∈据/span>(据/span>0.据/span>那据/span>2据/span>π据/span>)据/span>那据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>>据/span>0.据/span>。因此它是凹陷的。据/li>
- 为了据span class="katex">
X据/span>∈据/span>(据/span>2据/span>-据/span>π据/span>那据/span>0.据/span>)据/span>那据span class="katex">
F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>据据/span>0.据/span>。因此它是凹陷的。据/li>
结果:据B.r>我们有以下图:据/li>
切线据/span>
给予据span class="katex">
F据/span>(据/span>X据/span>)据/span>=据/span>晒黑据/span>-据/span>1据/span>1据/span>-据/span>X据/span>2据/span>2据/span>X据/span>,绘图图形据span class="katex">
y据/span>=据/span>F据/span>(据/span>X据/span>)据/span>。据/p>
自从据/p>
D.据/span>X据/span>D.据/span>(据/span>晒黑据/span>-据/span>1据/span>你据/span>)据/span>=据/span>1据/span>+据/span>你据/span>2据/span>1据/span>那据/span>
我们可以将给定的等式区分开:据/p>
D.据/span>X据/span>D.据/span>y据/span>=据/span>D.据/span>X据/span>D.据/span>(据/span>晒黑据/span>-据/span>1据/span>1据/span>-据/span>X据/span>2据/span>2据/span>X据/span>)据/span>=据/span>1据/span>+据/span>(据/span>1据/span>-据/span>X据/span>2据/span>2据/span>X据/span>)据/span>2据/span>1据/span>⋅据/span>D.据/span>X据/span>D.据/span>(据/span>1据/span>-据/span>X据/span>2据/span>2据/span>X据/span>)据/span>=据/span>1据/span>+据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>2据/span>4.据/span>X据/span>2据/span>1据/span>⋅据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>2据/span>2据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>-据/span>2据/span>X据/span>(据/span>-据/span>2据/span>X据/span>)据/span>=据/span>(据/span>1据/span>-据/span>X据/span>2据/span>)据/span>2据/span>+据/span>4.据/span>X据/span>2据/span>2据/span>-据/span>2据/span>X据/span>2据/span>+据/span>4.据/span>X据/span>2据/span>=据/span>(据/span>1据/span>+据/span>X据/span>2据/span>)据/span>2据/span>2据/span>(据/span>1据/span>+据/span>X据/span>2据/span>)据/span>=据/span>1据/span>+据/span>X据/span>2据/span>2据/span>。据/span>
再次差异化,我们得到了据/p>
D.据/span>X据/span>2据/span>D.据/span>2据/span>y据/span>=据/span>(据/span>1据/span>+据/span>X据/span>2据/span>)据/span>2据/span>-据/span>4.据/span>X据/span>。据/span>
arctan.据/span>
第二阶衍生物测试:据/strong>
一种据strong>第二衍生物测试据/strong>是用于确定静止点的测试据span class="katex">
F据/span>'据/span>(据/span>X据/span>)据/span>=据/span>0.据/span>是局部最大值或最小值。据/p>
执行测试:据/p>
如果据/span>F据/span>'据/span>'据/span>(据/span>X据/span>)据/span>⎩据/span>⎪据/span>⎨据/span>⎪据/span>⎧据/span>>据/span>0.据/span>那据/span>然后据/span>F据/span>有当地的最小值据/span>X据/span>。据/span>=据/span>0.据/span>那据/span>然后它是一个可能的拐点。据/span>据据/span>0.据/span>那据/span>然后据/span>F据/span>有当地的最大值据/span>X据/span>。据/span>