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<c val="#800000">Units in Functions and Operators</c>
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<phrase>units</phrase>
<phrase>type checking</phrase>
<phrase>error messages</phrase>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad 12 introduced static unit and type checking. All regions are unit balanced and checked for type (vector, scalar, function, and so on) consistency before they execute. Checking before evaluation offers gains in performance and consistency of operation. Doing so can cause some differences in your worksheets from earlier versions of Mathcad.</p>
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<p style="Header 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Functions That Don't Unit Balance Throw an Error When Defined</p>
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<b>MC11:</b> Functions return unit errors when evaluated.</p>
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<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC12 and higher:</b> Functions return unit errors when defined or evaluated.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You may notice that functions must unit balance when they are defined, not when they are executed. Click on the definition below to see the error message.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Because Mathcad no longer checks units while processing a calculation, you can not define a program or function that depends on a numeric value to determine the units of the output. The units in the following definition cannot be balanced without running it.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In some cases, the program may appear correct when you enter it, but returns an error when executed.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here, the problem arises from the omission of an <b>otherwise</b> from the <b>if</b> block. There are conceivable values of <b>adj</b> (for example, 2.5) that "fall through" the definition of factor, Mathcad assumes the definition of factor comes from the previous definition — in this case, <b>Vol</b>. This results in an inconsistency in units of the return value: if an acceptable value of <b>adj</b> is supplied, the program returns units of m<sup>3</sup>; if the value of <b>adj</b> causes the definition of factor to fail, the program returns units of m<sup>6</sup>.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">More information on the use of otherwise to avoid definition errors in programs can be found on the <b>
<u>
<link href=".\..\..\Programming\condtnl.xmcd" title="Links to section" popup="false">Conditional If Statements</link>
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</b> QuickSheet.</p>
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<p style="Header 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Static Unit Checking</p>
</text>
</region>
<region region-id="1434" left="6" top="870.75" width="321" height="57" align-x="6" align-y="882" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">With static unit checking, you also can not conditionally switch the output of a function based on the units of the arguments. This means you cannot use <b>on error</b> to trap unit errors or automatically add units if the arguments don't have them.</p>
</text>
<indexes>
<phrase>static unit checking</phrase>
<phrase>on error</phrase>
<phrase>unit error messages</phrase>
</indexes>
</region>
<region region-id="1435" left="36" top="942.75" width="234" height="36.75" align-x="112.5" align-y="954" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="The forms of the expression and its error handler must match.&#xA;The expression has the form: Length, &#xA;but the handler has the form: Length^2." optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">add_meters</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:program>
<ml:tryCatch>
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<ml:return>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:return>
</ml:tryCatch>
<ml:return>
<ml:id xml:space="preserve">x</ml:id>
</ml:return>
</ml:program>
</ml:define>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_handler		Value {Length 60000}	Value {Length 120000}	8	1	{Length 60000}	{Length 120000}	Length	Length^2</customValue>
</metadata>
<rendering item-idref="8"></rendering>
</region>
<region region-id="1392" left="6" top="996.75" width="333.75" height="28.5" align-x="6" align-y="1008" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The error messages are specific about which dimensions and types don't match. Click on each expression to see the message. </p>
</text>
</region>
<region region-id="1436" left="6" top="1044.75" width="321" height="28.5" align-x="6" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">For the same reasons, you cannot define a function that returns a different <i>type</i> of result depending on the value of its argument.</p>
</text>
</region>
<region region-id="1414" left="36" top="1092.75" width="121.5" height="36.75" align-x="68.25" align-y="1104" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="The forms of the possible return values must match.&#xA;The first has the form: ƒ(Unitless) → Unitless, &#xA;but the second has the form: Unitless." optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">f4</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:program>
<ml:ifThen>
<ml:apply>
<ml:lessThan></ml:lessThan>
<ml:id xml:space="preserve">x</ml:id>
<ml:real>0</ml:real>
</ml:apply>
<ml:id xml:space="preserve">sin</ml:id>
</ml:ifThen>
<ml:otherwise>
<ml:real>1</ml:real>
</ml:otherwise>
</ml:program>
</ml:define>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_consequent		(V {})-&gt;V {}	Value {}	4	1	-&gt;	Value	ƒ(Unitless) → Unitless	Unitless</customValue>
</metadata>
<rendering item-idref="9"></rendering>
</region>
<region region-id="1413" left="6" top="1140.75" width="339.75" height="28.5" align-x="6" align-y="1152" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You cannot mix functions and values, either in a function definition, or in matrix entries.</p>
</text>
</region>
<region region-id="1675" left="0" top="1182" width="6000" height="6" align-x="0" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1394" left="6" top="1184.25" width="241.5" height="12" align-x="6" align-y="1194" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Header 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Functions Valid Only for Unitless Arguments </p>
</text>
<indexes>
<phrase>functions, unitless arguments</phrase>
</indexes>
</region>
<region region-id="1477" left="6" top="1202.25" width="308.25" height="24" align-x="25.5" align-y="1212" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC11:</b> Functions can conditionally return results of different units depending on input values.</p>
</text>
</region>
<region region-id="1480" left="6" top="1232.25" width="308.25" height="24" align-x="25.5" align-y="1242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC12 and higher:</b> Functions must return units depending only on the units of the inputs.</p>
</text>
</region>
<region region-id="1016" left="6" top="1266.75" width="324" height="71.25" align-x="6" align-y="1278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Some functions may only be valid for unitless arguments in Mathcad 12 and higher versions that are valid for arguments with units in previous versions. For these functions, units on inputs cause the units of the output to vary based on the numerical values of one or more of the inputs, which doesn't correspond to a correctly unit-balanced equation. </p>
</text>
</region>
<region region-id="1015" left="36" top="1351.5" width="123.75" height="47.25" align-x="68.25" align-y="1368" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">f3</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:program>
<ml:ifThen>
<ml:apply>
<ml:lessThan></ml:lessThan>
<ml:id xml:space="preserve">x</ml:id>
<ml:real>0</ml:real>
</ml:apply>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">x</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:ifThen>
<ml:otherwise>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">x</ml:id>
<ml:real>3</ml:real>
</ml:apply>
</ml:otherwise>
</ml:program>
</ml:define>
</math>
<rendering item-idref="10"></rendering>
</region>
<region region-id="1397" left="6" top="1410.75" width="304.5" height="28.5" align-x="6" align-y="1422" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">These functions will not execute if you use expressions with units in the arguments. Click on each error to see the new error messages.</p>
</text>
</region>
<region region-id="943" left="36" top="1452.75" width="54" height="15.75" align-x="68.25" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">f3</ml:id>
<ml:real>2</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>8</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="11"></rendering>
</region>
<region region-id="1741" left="144" top="1452.75" width="50.25" height="14.25" align-x="144" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">works, but</p>
</text>
</region>
<region region-id="1748" left="210" top="1452.75" width="69" height="15.75" align-x="258" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value has units: Length, &#xA;but must have units: Unitless." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve" font="1">f3</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_dimensions	Unitless	Length</customValue>
</metadata>
<rendering item-idref="12"></rendering>
</region>
<region region-id="1734" left="318" top="1452.75" width="45.75" height="14.25" align-x="318" align-y="1464" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fails.</p>
</text>
</region>
<region region-id="945" left="36" top="1477.5" width="75.75" height="21" align-x="90.75" align-y="1494" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">pow</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="13"></rendering>
</region>
<region region-id="1057" left="36" top="1506.75" width="77.25" height="15.75" align-x="91.5" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">pow</ml:id>
<ml:sequence>
<ml:real>2</ml:real>
<ml:real>3</ml:real>
</ml:sequence>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>8</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="14"></rendering>
</region>
<region region-id="1740" left="144" top="1506.75" width="50.25" height="14.25" align-x="144" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">works, but</p>
</text>
</region>
<region region-id="1749" left="210" top="1506.75" width="92.25" height="15.75" align-x="281.25" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value has units: Length, &#xA;but must have units: Unitless." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">pow</ml:id>
<ml:sequence>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<ml:real>3</ml:real>
</ml:sequence>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_dimensions	Unitless	Length</customValue>
</metadata>
<rendering item-idref="15"></rendering>
</region>
<region region-id="1736" left="318" top="1506.75" width="47.25" height="14.25" align-x="318" align-y="1518" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fails.</p>
</text>
</region>
<region region-id="1439" left="36" top="1536.75" width="123" height="15.75" align-x="67.5" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">tu</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:tryCatch>
<ml:apply>
<ml:plus></ml:plus>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<ml:id xml:space="preserve">x</ml:id>
</ml:tryCatch>
</ml:define>
</math>
<rendering item-idref="16"></rendering>
</region>
<region region-id="1442" left="36" top="1566.75" width="103.5" height="15.75" align-x="83.25" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">tu</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>1</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>6.561679790026246</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="foot"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="17"></rendering>
</region>
<region region-id="1739" left="144" top="1566.75" width="49.5" height="14.25" align-x="144" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">works, but</p>
</text>
</region>
<region region-id="1750" left="210" top="1566.75" width="52.5" height="15.75" align-x="241.5" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value has units: Unitless, &#xA;but must have units: Length." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">tu</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_dimensions	Length	Unitless</customValue>
</metadata>
<rendering item-idref="18"></rendering>
</region>
<region region-id="1738" left="318" top="1566.75" width="43.5" height="14.25" align-x="318" align-y="1578" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fails.</p>
</text>
</region>
<region region-id="1494" left="0" top="1596" width="6000" height="6" align-x="0" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1495" left="6" top="1602.75" width="341.25" height="71.25" align-x="6" align-y="1614" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">As a general rule, for a function to be valid in Mathcad 12 through Mathcad 14, the units of the result must depend only on the units of the arguments to the function. Trace Error does not work in these cases, because the error is caused by evaluation of the function with units in the arguments, not by the function definition itself.</p>
</text>
<indexes>
<phrase>Trace Error</phrase>
</indexes>
</region>
<region region-id="1400" left="6" top="1688.25" width="281.25" height="24" align-x="6" align-y="1698" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Header 2" margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit">Exponentiation and Differentiation Operators need Explicit Power and Order with Arguments with Units</p>
</text>
<indexes>
<phrase>exponentiation operators</phrase>
<phrase>differentiation operators</phrase>
</indexes>
</region>
<region region-id="1499" left="6" top="1724.25" width="329.25" height="12" align-x="25.5" align-y="1734" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC11:</b> Values with units can be raised to a variable power.</p>
</text>
</region>
<region region-id="1500" left="6" top="1742.25" width="308.25" height="24" align-x="25.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC12 through MC14:</b> Values with units can be raised to an explicit numeric power.</p>
</text>
</region>
<region region-id="1401" left="6" top="1776.75" width="333.75" height="57" align-x="6" align-y="1788" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To unit balance an exponent or derivative equation before evaluation, the units of the result must depend only on the units of the arguments, not on the numeric value of the arguments. Explicit values are needed to calculate the powers of a quantity with units.</p>
</text>
</region>
<region region-id="966" left="36" top="1842.75" width="35.25" height="15.75" align-x="54.75" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">ex</ml:id>
<ml:real>3</ml:real>
</ml:define>
</math>
<rendering item-idref="19"></rendering>
</region>
<region region-id="1403" left="126" top="1842.75" width="71.25" height="15.75" align-x="158.25" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">anum</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="20"></rendering>
</region>
<region region-id="1086" left="36" top="1867.5" width="75.75" height="21" align-x="72.75" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">anum</ml:id>
<ml:real>3</ml:real>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>8</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="second" power-numerator="3"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="21"></rendering>
</region>
<region region-id="990" left="132" top="1872.75" width="24.75" height="14.25" align-x="132" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">but</p>
</text>
</region>
<region region-id="988" left="168" top="1867.5" width="62.25" height="21" align-x="209.25" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="The exponentiation operator requires a constant &#xA;exponent when applied to a value with units." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">anum</ml:id>
<ml:id xml:space="preserve">ex</ml:id>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_exponent	Unitless	Time</customValue>
</metadata>
<rendering item-idref="22"></rendering>
</region>
<region region-id="992" left="240" top="1872.75" width="78.75" height="14.25" align-x="240" align-y="1884" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">gives an error</p>
</text>
</region>
<region region-id="1407" left="6" top="1902.75" width="321" height="42.75" align-x="6" align-y="1914" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">This error message says that you can only raise a unitless value to a variable power. You also need to provide explicit values for the order of an nth-order derivative of a dimensioned value.</p>
</text>
</region>
<region region-id="1137" left="36" top="1960.5" width="127.5" height="33.75" align-x="79.5" align-y="1980" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">pos</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">t1</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>3</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<ml:apply>
<ml:id xml:space="preserve">sin</ml:id>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">t1</ml:id>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="23"></rendering>
</region>
<region region-id="1141" left="36" top="2004.75" width="40.5" height="15.75" align-x="60" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">ord</ml:id>
<ml:real>2</ml:real>
</ml:define>
</math>
<rendering item-idref="24"></rendering>
</region>
<region region-id="1145" left="114" top="2004.75" width="56.25" height="15.75" align-x="131.25" align-y="2016" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">t1</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="25"></rendering>
</region>
<region region-id="1139" left="36" top="2031.75" width="132" height="45.75" align-x="104.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:derivative></ml:derivative>
<ml:lambda>
<ml:boundVars>
<ml:id xml:space="preserve">t1</ml:id>
</ml:boundVars>
<ml:apply>
<ml:id xml:space="preserve">pos</ml:id>
<ml:id xml:space="preserve">t1</ml:id>
</ml:apply>
</ml:lambda>
<ml:degree>
<ml:real>2</ml:real>
</ml:degree>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>-8.949777823084565</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="foot"></unitReference>
<unitReference unit="second" power-numerator="-2"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="26"></rendering>
</region>
<region region-id="1154" left="162" top="2046.75" width="24.75" height="14.25" align-x="162" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">but</p>
</text>
</region>
<region region-id="1467" left="192" top="2031.75" width="98.25" height="44.25" align-x="269.25" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="The order of the derivative must be a constant &#xA;between 1 and 5 when this operator is applied &#xA;to an expression with units." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:derivative></ml:derivative>
<ml:lambda>
<ml:boundVars>
<ml:id xml:space="preserve">t1</ml:id>
</ml:boundVars>
<ml:apply>
<ml:id xml:space="preserve">pos</ml:id>
<ml:id xml:space="preserve">t1</ml:id>
</ml:apply>
</ml:lambda>
<ml:degree>
<ml:id xml:space="preserve">ord</ml:id>
</ml:degree>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_derivative	(Unitless) ? any1	(Time) ? Length</customValue>
</metadata>
<rendering item-idref="27"></rendering>
</region>
<region region-id="1725" left="288" top="2046.75" width="72.75" height="14.25" align-x="288" align-y="2058" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">gives an error</p>
</text>
</region>
<region region-id="1409" left="6" top="2088.75" width="360" height="28.5" align-x="6" align-y="2100" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The error message is telling you that you can only take a variable-order derivative with respect to a unitless argument of a function. </p>
</text>
</region>
<region region-id="1552" left="0" top="2130" width="6000" height="6" align-x="0" align-y="2130" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1505" left="6" top="2138.25" width="371.25" height="12" align-x="6" align-y="2148" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Header 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">2D Plots Require Matching Units in Axes and Markers</p>
</text>
<indexes>
<phrase>2D plots, matching units in axes</phrase>
</indexes>
</region>
<region region-id="1506" left="6" top="2162.25" width="308.25" height="24" align-x="25.5" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC11 and MC12:</b> Axis limits and markers in 2D plots can have mismatched units.</p>
</text>
</region>
<region region-id="1507" left="6" top="2192.25" width="308.25" height="24" align-x="25.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="comment" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>MC13 and MC14:</b> Axis limits and markers in 2D plots are required to have the same units.</p>
</text>
</region>
<region region-id="1508" left="6" top="2226.75" width="341.25" height="71.25" align-x="6" align-y="2238" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To scale plots correctly in the current worksheet system base units, a plot must use consistent dimensions for values that determine the scale of an axis. Axis limits and markers on a plot, if they use units, must have consistent units. You may find that plots that display in Mathcad 11 and 12 return unit errors in Mathcad 13 and 14.</p>
</text>
</region>
<region region-id="1582" left="48" top="2316.75" width="57" height="20.25" align-x="67.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve" subscript="p">x</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2.3</ml:real>
<ml:id xml:space="preserve">s</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="28"></rendering>
</region>
<region region-id="1587" left="126" top="2311.5" width="144" height="25.5" align-x="153.75" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">f</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:neg></ml:neg>
<ml:apply>
<ml:pow></ml:pow>
<ml:parens>
<ml:apply>
<ml:minus></ml:minus>
<ml:id xml:space="preserve" subscript="p">x</ml:id>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
</ml:parens>
<ml:real>2</ml:real>
</ml:apply>
</ml:apply>
<ml:id xml:space="preserve">g</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>40</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="29"></rendering>
</region>
<region region-id="1595" left="300" top="2316.75" width="48.75" height="15.75" align-x="310.5" align-y="2328" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">i</ml:id>
<ml:range>
<ml:real>0</ml:real>
<ml:real>10</ml:real>
</ml:range>
</ml:define>
</math>
<rendering item-idref="30"></rendering>
</region>
<region region-id="1561" left="48" top="2338.5" width="57.75" height="33.75" align-x="63.75" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">i</ml:id>
<ml:real>2</ml:real>
</ml:apply>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="31"></rendering>
</region>
<region region-id="1562" left="132" top="2346.75" width="49.5" height="17.25" align-x="147.75" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">y</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:id xml:space="preserve">f</ml:id>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="32"></rendering>
</region>
<region region-id="1589" left="204" top="2346.75" width="68.25" height="20.25" align-x="234.75" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">peak</ml:id>
<ml:apply>
<ml:id xml:space="preserve">f</ml:id>
<ml:id xml:space="preserve" subscript="p">x</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="33"></rendering>
</region>
<region region-id="1596" left="300" top="2346.75" width="98.25" height="15.75" align-x="330" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">peak</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>131.23359580052494</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="foot"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="34"></rendering>
</region>
<region region-id="1578" left="42" top="2382" width="200.25" height="109.5" align-x="42" align-y="2382" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="35"></plot>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_array_element	Unitless	Length</customValue>
</metadata>
<rendering item-idref="36"></rendering>
</region>
<region region-id="1579" left="6" top="2514.75" width="324" height="57" align-x="6" align-y="2526" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The problem with the plot above is that the y-axis limits are set to unitless values, 150 and -150, and the marker has units of length. You can either add units to the plot limits or divide the units from the marker to display the plot.</p>
</text>
</region>
<region region-id="1575" left="30" top="2586" width="237.75" height="136.5" align-x="30" align-y="2586" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<plot disable-calc="false" item-idref="37"></plot>
<rendering item-idref="38"></rendering>
</region>
<region region-id="1601" left="0" top="2724" width="6000" height="6" align-x="0" align-y="2724" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1600" left="6" top="2730.75" width="339" height="71.25" align-x="6" align-y="2742" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The units on markers and axis limits do not have to match the units on the axis arguments, since multiple arguments can have different units in Mathcad plots. We recommend that in these cases you normalize all values to be unitless, since this is the most correct and consistent way to compare values, which allows you to specifically define the scaling.</p>
</text>
</region>
<region region-id="1608" left="6" top="2820.75" width="345.75" height="71.25" align-x="6" align-y="2832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In the case where all axis arguments have the same units, we recommend that you make units on plot limits and markers match the argument units, for easy interpretation of scaling. Plots do not follow unit simplification rules, so all units will be reduced to the base dimensions of the system. This automatic scaling can result in surprising values on numbered plots. For example,</p>
</text>
</region>
<region region-id="1648" left="36" top="2906.25" width="103.5" height="12" align-x="36" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="h5" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Default evaluation</p>
</text>
</region>
<region region-id="1661" left="156" top="2906.25" width="112.5" height="12" align-x="171.75" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="h5" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Default graph scaling</p>
</text>
</region>
<region region-id="1656" left="288" top="2904.75" width="54.75" height="17.25" align-x="309.75" align-y="2916" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:apply>
<ml:indexer></ml:indexer>
<ml:id xml:space="preserve">y2</ml:id>
<ml:id xml:space="preserve">i</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">i</ml:id>
<ml:id xml:space="preserve">lbf</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="39"></rendering>
</region>
<region region-id="1652" left="36" top="2934.75" width="99" height="93.75" align-x="54.75" align-y="2946" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">y2</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:matrix rows="11" cols="1">
<ml:real>0</ml:real>
<ml:real>1</ml:real>
<ml:real>2</ml:real>
<ml:real>3</ml:real>
<ml:real>4</ml:real>
<ml:real>5</ml:real>
<ml:real>6</ml:real>
<ml:real>7</ml:real>
<ml:real>8</ml:real>
<ml:real>9</ml:real>
<ml:real>10</ml:real>
</ml:matrix>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="pound_force"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
<resultFormat>
<table item-idref="40"></table>
</resultFormat>
</math>
<rendering item-idref="41"></rendering>
</region>
<region region-id="1660" left="156" top="2926.5" width="132" height="102" align-x="174.75" align-y="2946" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">y2</ml:id>
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<ml:real>225.21833989501309</ml:real>
<ml:real>257.39238845144354</ml:real>
<ml:real>289.56643700787396</ml:real>
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<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Although the default simplification for force in the US unit system is lbf, graphs will simplify to the 7 base units.</p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">In this case, graphs have default scaling that is approximately 32 times larger than the default simplified unit.</p>
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