lecture14Student(1).pdf - Lecture 14 Unit 11 Integration Outline 1 Indefinite integration Rules for indefinite integration 2 Definite integration 3

lecture14Student(1).pdf - Lecture 14 Unit 11 Integration...

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Lecture 14, Unit 11: Integration 22/4/2016
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Outline 1 Indefinite integration Rules for indefinite integration 2 Definite integration 3 Economic Applications Derive TC from MC Derive TR from MR Consumer surplus Producer surplus 4 Other applications 5 Summary
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Objectives Use the rules of integration to find integrals Find total functions from marginal functions Find consumer surplus Find producer surplus
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Literature Renshaw, ch. 18
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Integration Indefinite integration - reverses differentiation, in the same way the multiplication reverses division. Also known as the anti-derivative Definite integration - allows us to calculate the area under a function/curve (area between x-axis and curve)
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Indefinite integral, p. 554-61 If d dx F ( x ) = f ( x ), then R f ( x ) d x = F ( x ) + c R = integral sign, f ( x ) is the integrand, c = constant of integration, dx indicates that we integrate w.r.t. x Note the R f ( x ) just means integral sign f(x), while R f ( x ) d x means that we have to integrate f(x) w.r.t. x. Notation is important here. The relationship between indefinite integral and derivative: d dx Z f ( x ) d x = f ( x ) The derivative of the indefinite integral = the integrand Z f ( x ) d x = F ( x ) + c The indefinite integral of the integrand is the indefinite integral
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Power rule Recall that d dx ( ax n ) = nax n - 1 Now, note that to reverse differentiation: Z x n d x = x n +1 n + 1 for n 6 = - 1 Example: find R x 0 . 5 d x Z x 0 . 5 d x = x 0 . 5+1 0 . 5 + 1 + c = x 1 . 5 1 . 5 = 2 3 x 1 . 5 + c Note what happens when we differentiate this function: d dx 2 3 x 1 . 5 + c = x 0 . 5 Also note that R d x = R 1 d x = R x 0 d x = x 0+1 0+1 + c = x + c
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Multiplicative constant rule Recall that d dx Ax = A , d dx Af ( x ) = Af 0 ( x ) To reverse this Z Af ( x ) d x = A Z f ( x ) d x = AF ( x ) + c Example: find R 100 x 3 d x Z 100 x 3 d x = 100 Z x 3 d x = 100 x 4 4 + c = 25 x 4 + c Derivative of this function is ...
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Sum/difference rule Recall that d dx ( f ( x ) ± g ( x )) = f 0 ( x ) ± g 0 ( x ) To reverse this Z [ f ( x ) ± g ( x )] d x = Z f ( x ) d x ± Z g ( x ) d x = F ( x ) ± G ( x )+ c Example: find R (5 x 3 + 3 x 2 ) d x Z (5 x 3 + 3 x 2 ) d x = Z 5 x 3 d x + Z 3 x 2 d x = 5 Z x 3 d x + 3 Z x 2 d x + c = 5 x 4 4 + 3 x 3 3 + c = 1 . 25 x 4 + x 3 + c Derivative of this function is ...
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Function of a function rule This rule reverses the chain rule of differentiation Recall that the chain rule states that d dx ( f ( x )) n = nf ( x ) n - 1 df ( x ) dx = nf ( x ) n - 1
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