Fact Sheet for Final

Fact Sheet for Final - FACT SHEET FOR 20C FINAL EXAM...

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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 F z . 2. Integrals Over Boxes (2D) In 2D we integrate a function f ( x, y ) over the box a x b , c y d : I = b a d c f ( x, y ) dydx .
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