示例：参数解

函数 > 求解和最优化 > 示例：参数解

示例：参数解

使用 root 函数或求解命令块来定义函数，用于计算从属于未知参数的方程或方程组。这将求解整个方程组。

Root 函数

求解不同参数值 a 的方程 f1=f2。

1. 定义两个函数。

2. 定义新函数 S 以求解不同 a 值的方程。

3. 定义初始估值。

4. 定义参数 a 的范围并求解各个不同值的方程。

求解命令块：摩擦系数

求出给定 e (管道粗糙度)、D (管道直径) 和 R (雷诺数) 值的摩擦系数 f。

1. 定义估值。

2. 编写求解命令块以定义函数来计算摩擦系数。

<region id="ID0E64I5" top="1114.96062992126" left="56.6929133858268" width="265.6" height="112.8" xmlns="http://schemas.mathsoft.com/worksheet50">
  <solveblock>
    <regions>
      <region id="" top="48" left="24" solve-block-category="constraint">
        <math disable-calc="false">
          <apply xmlns="http://schemas.mathsoft.com/math50">
            <equal />
            <apply>
              <div />
              <real>1</real>
              <apply>
                <nthRoot />
                <placeholder />
                <id>f</id>
              </apply>
            </apply>
            <apply>
              <mult />
              <real>-2</real>
              <apply>
                <id>log</id>
                <apply>
                  <plus />
                  <apply>
                    <div />
                    <apply>
                      <mult />
                      <id>ε</id>
                      <apply>
                        <pow />
                        <id>D</id>
                        <real>-1</real>
                      </apply>
                    </apply>
                    <real>3.7</real>
                  </apply>
                  <apply>
                    <div />
                    <real>2.51</real>
                    <apply>
                      <mult />
                      <id>R</id>
                      <apply>
                        <nthRoot />
                        <placeholder />
                        <id>f</id>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </math>
      </region>
      <region id="" top="96" left="24" solve-block-category="solver">
        <math disable-calc="false">
          <define xmlns="http://schemas.mathsoft.com/math50">
            <function>
              <id label="Function">FricFac</id>
              <boundVars>
                <id label="Variable">ε</id>
                <id label="Variable">D</id>
                <id label="Variable">R</id>
              </boundVars>
            </function>
            <apply>
              <id label="Keyword">find</id>
              <id>f</id>
            </apply>
          </define>
        </math>
      </region>
    </regions>
  </solveblock>
</region>

参数 e、D 和 R 尚未定义。每次计算函数 FricFac 时，都可以为这些参数指定新值，求解命令块将会被再次调用。

3. 计算不同雷诺数的摩擦系数。

<region id="ID0EKCJ5" top="1568.50393700787" left="56.6929133858268" width="547.659842519685" height="302.362204724409" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace>
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <axes>
        <xAxis legend-position="PlotBoundaryBottom" rank="1">
          <axisLine position="bottom" positionticmark="-1" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <id label="Variable">R</id>
                  <id>i</id>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis legend-position="PlotBoundaryLeft" rank="1">
          <axisLine position="left" positionticmark="-1" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <id label="Variable">ff</id>
                  <id>i</id>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

求解命令块：圆和直线相交

1. 求出随着圆半径的变化，圆和直线 y = 2 - x 相交的位置。

<region id="ID0ENHJ5" top="2229.92125984252" left="56.6929133858268" width="430.383567271483" height="396.506675795958" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="true">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace>
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace>
          <traceStyle color="#FF9BBB59" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace>
          <traceStyle color="#FF8DB3E2" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
      </traces>
      <axes>
        <xAxis legend-position="PlotBoundaryBottom" rank="1">
          <axisLine position="origin" positionticmark="4" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">X</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis legend-position="PlotBoundaryLeft" rank="1">
          <axisLine position="origin" positionticmark="4" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">TopCir</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">BotCir</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">Line</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

2. 编写求解命令块来定义函数，以返回直线 y = 2 - x 和半径为 R 的圆之间的交点的坐标。

3. 使用新定义的 Z 函数求出 R=3 与 x=1 及 y=1 估值的交点坐标。

4. 绘制交点的图像。

<region id="ID0ERNJ5" top="3269.29133858268" left="56.6929133858268" width="433.583567271483" height="486.95652173913" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot origin-positioning="false">
    <xyPlot>
      <title class="- topic/title " wwtype:type="Paragraph" xmlns:wwtype="urn:WebWorks-Type-Schema" />
      <legend />
      <traces>
        <trace>
          <traceStyle color="#FF8DB3E2" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace>
          <traceStyle color="#FF9BBB59" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace>
          <traceStyle color="#FF00008B" symbol="none" line-weight="1" line-style="Solid">lines</traceStyle>
        </trace>
        <trace>
          <traceStyle color="#FF000000" symbol="rectangle" line-weight="1" line-style="None">lines</traceStyle>
        </trace>
      </traces>
      <axes>
        <xAxis legend-position="PlotBoundaryBottom" rank="1">
          <axisLine position="ticknumberlock" positionticmark="4" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">X</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">X</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">X</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <apply>
                    <id>Z</id>
                    <real>3</real>
                  </apply>
                  <real>1</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis legend-position="PlotBoundaryLeft" rank="1">
          <axisLine position="ticknumberlock" positionticmark="4" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat />
          <plotEquations>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">TopCir</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">BotCir</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <id xmlns="http://schemas.mathsoft.com/math50">Line</id>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
            <plotEquation>
              <math>
                <apply xmlns="http://schemas.mathsoft.com/math50">
                  <indexer />
                  <apply>
                    <id>Z</id>
                    <real>3</real>
                  </apply>
                  <real>0</real>
                </apply>
              </math>
              <math>
                <placeholder xmlns="http://schemas.mathsoft.com/math50" />
              </math>
            </plotEquation>
          </plotEquations>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <real xmlns="http://schemas.mathsoft.com/math50">5</real>
            </endValue>
          </xyDomain>
        </yAxis>
      </axes>
    </xyPlot>
  </plot>
</region>

5. 使用新定义的 Z 函数求出 5 个不同半径 R 值的交点坐标。

也可以通过定义 Z(R, x, y)，根据估值来参数化以上求解命令块。这将可以使用单个求解命令块设置为每个半径计算两个不同的解。