π π Can really be treated as a π π quotient? Ta m a r Av i n e r i No r t h C a r o l i n a S c h o o l O f S c i e n ce A n d M a t h e m a t i c s Te a c h i n g C o n te m p o ra r y M a t h e m a t i c s C o n f e r e n ce Ja n u a r y 2 7 - 2 8 , 2 0 17 Solving a Differential Equation Solving a Differential Equation Mathematical Nuance β’ Is ππ¦ an operator, a number, a function, a derivativeβ¦? β’ Whatβs the difference between Ξπ¦, ππ¦, β’ If β«β π’ Χ¬β¬ β’ Since ππ’ ππ₯ ππ₯ ππ¦ ππ₯ = π ππ¦ , ? ππ₯ ππ₯ = β«π’ππ’ Χ¬β¬, then can we say that ππ’ ππ₯ ππ₯ ππ¦ ππ’ , then why canβt we say that the ππ’ ππ₯ ππ¦ ππ¦ ππ’ = ππ₯ ππ’ ππ₯ = ππ’? ππ’β² π cancel? Forum Post http://mathforum.org/library/drmath/view/65462.html Is it worth class discussion? http://mathforum.org/library/drmath/view/65462.html Ξπ¦ vs. ππ¦ πβ² π π₯0 + Ξπ₯ β π(π₯0 ) π₯0 β Ξπ₯ when Ξπ₯ is small. So, π π₯0 + Ξπ₯ β π π₯0 β π β² π₯0 Ξπ₯ = Ξπ¦. Ξπ¦ vs. ππ¦ Consider π π₯ = π₯ 2 . Compare ππ¦ and Ξπ¦ when π₯ = 1, ππ₯ = 0.01, and Ξπ₯ = 0.01. π β² π₯ = 2π₯ ππ¦ = π β² π₯ ππ₯ When π₯ = 1, we have ππ¦ = π β² 1 0.01 = 2 0.01 = 0.02. Ξπ¦ = π π₯ + Ξπ₯ β π π₯ When π₯ = 1, π 1.01 β π 1 = 1.01 2 β 12 = 0.0201. Definition of Differentials Let π¦ = π(π₯) represent a function that is differentiable in an open interval containing π₯. The differential of π (denoted by ππ₯) is any nonzero real number. The differential of π (denoted by ππ¦) is π π = πβ² π π π. Calculus, by Larson, Hostetler, Edwards, 2002 Differential Form of the Product Rule Consider differentiable functions π’(π₯) and π£(π₯). So, ππ’ = π’β² ππ₯ and ππ£ = π£ β² ππ₯. π π’π£ = π ππ₯ π’π£ ππ₯ Differential of uv = π’π£ β² + π£π’β² ππ₯ Product Rule = π’π£ β² ππ₯ + π£π’β² ππ₯ = π’ππ£ + π£ππ’ Calculus, by Larson, Hostetler, Edwards, 2002 U-substitution U-substitution Integration by Parts Integration by Parts Integration by Parts ΰΆ± π₯π π₯ ππ₯ 1) Let π’ = π₯ and ππ£ ππ₯ = π π₯ . So, ππ’ ππ₯ = 1 and π£ = π π₯ ? OR 2) Let π’ = π₯ and ππ£ = π π₯ ππ₯. So, ππ’ = ππ₯ and π£ = π π₯ ? Separation of Variables Opinion Blog βMost textbooks and classes gloss over the fact that dy/dx is not a ratio of two quantities, and treat the individual terms as though they can be multiplied, divided and canceled at will. Some authors do at least point out the problem, but usually to say βdy/dx is not a ratio, but it can be helpful on occasions to treat it as if it were.β It's never very clear what these occasions are. Sometimes you'll come across statements such as βThis procedure is justified by the chain ruleβ or βThis procedure is justified by the Fundamental Theorem of Calculus.β Maybe that's true, but how is it justified? You'll sometimes see expressions in dx or dy explained away with hand-waving expressions like βformal formβ or βinfinitesimal formβ. But what do those terms really mean?β -Kevin Boone http://http://www.kevinboone.net/separation_variables.html Continued Opinion βNevertheless, it seems to me that explaining the methodology of separation of variables and integration by substitution as a short-cut to the application of the chain rule is more likely to be understood by students, than a highly technical discussion of the meaning of a differential.β -Kevin Boone http://http://www.kevinboone.net/separation_variables.html Leibniz Notation βThe beauty of this notation is that it provides an easy way to remember several important calculus formulas by making it seem as though the formulas were derived from algebraic manipulations of differentials.β -Larson, Hostetler, and Edwards, 2002, p. 231 Questions/Discussion Tamar Avineri North Carolina School of Science and Mathematics [email protected]
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