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Unformatted text preview: FACT SHEET FOR 20C FINAL EXAM Abstract. Here are some formulas that you should know for the second exam. Please bring a copy of this sheet to the test. Copies will not be included with the exams. Please do not write extra formulas on this sheet. 1. Chain Rules and Implicit Differentiation • (Chain Rule I) Given f ( x, y ), x ( t ), and y ( t ), we can form a function of t alone by composing h ( t ) = f ( x ( t ) , y ( t ) ) . Then: dh dt = ∂ x f dx dt + ∂ y f dy dt . • (Chain Rule II) Given f ( x, y ), x ( u, v ), and y ( u, v ), we can form a new function of ( u, v ) by composing h ( u, v ) = f ( x ( u, v ) , y ( u, v ) ) . Then: ∂h ∂u = ∂ x f ∂x ∂u + ∂ y f ∂y ∂u , ∂h ∂v = ∂ x f ∂x ∂v + ∂ y f ∂y ∂v . • (Implicit Differentiation) Given a relation F ( x, y, z ) = 0, this defines an implicit function z = f ( x, y ). One can then compute the partial derivatives of z = f without solving for z via the formulas: ∂z ∂x =- F x F z , ∂z ∂y =- F y...
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This note was uploaded on 12/08/2011 for the course MATH 20C 20C taught by Professor Boonyeap during the Fall '07 term at UCSD.
- Fall '07