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# 10.05 - [Hand out time sheets Questions on chapter 10 If...

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[Hand out time sheets] Questions on chapter 10? If you have questions on sections 11.1 or 11.2, please send me email, so I can prepare explanations to present in class. Section 11.3: Partial derivatives Main ideas: Definition of partial derivatives Geometrical meaning of partial derivatives as slopes of traces Higher partial derivatives How to estimate partial derivatives How to compute partial derivatives Clairaut’s Theorem Partial differential equations Discuss definitions 1, 2, 3, and 4 on page 609, and the variant notations f 1 ( x , y ) and f 2 ( x , y ). Given a function f ( x , y ), we can write its graph as {( x , y , z ): x , y , z in R , z = f ( x , y )}

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or {( x , y , f ( x , y )): x , y in R }. We can intersect this surface with the plane y = b , obtaining the trace {( x , b , f ( x , b )): x in R }. Write this trace as {( x , b , g ( x )): x in R } where g ( x ) is defined as f ( x , b ). If we look at this trace as a 2-dimensional curve in the plane y = b , we can look at the tangent to the curve at the point P = ( a , b , c ) = ( a , b , g ( a )). The slope of this tangent line is g ( a ), which was Stewart’s definition of f 1 ( a , b
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