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Unformatted text preview: Lecture Notes for Week 6 Differentiation Derivatives as Rates of Change Let p = f ( q ) the demand curve for a certain product. Common experience tell us that modifying the demand leads to a change in price. Let A = ( q A , p A ) be a point on the curve p = f ( q ) . Suppose the demand rose from q A to q S = q A + q products. Then the price is modified to p S = p A + p. The slope m AS of the line AS is m AS = p S- p A q S- q A = ( p A + p )- p A ( q A + p )- q A = p q m AS = p q So, the change is price as given by p = m AS q If we would know the slope of the line AS as S is closer and closer to A, we could say how the price is instantaneously changing when the price is actually q A . But, as S is closer and closer to A, the line AS as S becomes in fact the tangent line on the curve p = f ( q ) at A. The instantaneous rate of change of the price at A is, by definition, the slope of the tangent line on the curve p = f ( q ) at A and it is given by: p S- p A q S- q A where S is very-very close to A Note that S can never be taken to be exactly A, for then the slope becomes , which makes no sense (there is no division by zero!) In math notation, we write S is getting close and closer to A as lim S A and read limit as S tends to A . With this notation, the slope of the tangent line at p = f ( q ) at A (or the rate of change of the price at A ) is written as: m A = lim S A p S- p A q S- q A and reads m is the limit of p S- p A q S- q A as S tends to A. Definition Given a curve y = f ( x ) , the derivative of f at a point A := ( x , y ) on the curve is f ( x ) := m A = lim S A y S- y A x S- x A . In other words, the derivative of f at a point A := ( x , y ) is the rate of change of f at A. Equivalently, the derivative of f at a point A := ( x , y ) is the slope of the tangent line to y = f ( x ) at A. We also use the notations: f ( x ) = f ( x ) x = x = dy dx x = x Question The total cost function for a product is c ( q ) = 3 q 3- 5 q . What is the rate of change of the cost when q = 2?...
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This note was uploaded on 12/12/2011 for the course ECON 101 taught by Professor Bi during the Spring '11 term at York University.
- Spring '11