基于图像变换的拉格朗日中值定理辅助函数构建方法

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中图分类号：0172.1 文献标识码：A

Abstract：Lagrange's Mean Value Theoremisacrucial toolforanalyzing therelationshipbetwen functionsandderivatives， helpingstudentsunderstandtheconnectionbetwnderivativesandterateofchangeoffunctions.Inteachingpracticeinstuctors primarilyemployvariousmethods，suchastheconstantkmethod，thedeterminantmethod，andtherotationofcoordinateaes，to construct auxiliaryfunctions，therebyhelping students tounderstandandapplyLagrange's Mean Value Theoremmoredeply.This paperproposesanovelmethodforconstructingauxiliaryfunctionsfromtheperspectiveoffunctiongraph transformation.Byanaly zingthetranslationandsymmetrypropertiesofthefunction's graph，thisnewmethodallowsauxiliaryfunctions tobederiveddirectlyfromthevisualrepresentation.This more intuitiveapproach helpsstudents clarifytheirreasoning inconstructingauxiliaryfunctions，facilitating theproof ofLagrange'sMean Value Theoremandmaking itmorepracticallymeaningfulandcomprehensible inan educational context.

Keywords：Lagrange's Mean Value Theorem;AuxiliaryFunctions;Graph Transformation;Mathematics Education

1引言

拉格朗日中值定理是微积分学中的重要定理，掌握其内容和证明过程对学习导数的应用有重要的作用。（剩余4285字）