MTH-140-Lecture-14-Winter-2012 - Ryerson University MTH 140 Winter 2012 Lecture 14 Antiderivatives Areas The content for this lecture is in sections 4.9

# MTH-140-Lecture-14-Winter-2012 - Ryerson University MTH 140...

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Ryerson University MTH 140, Winter 2012 Lecture 14: Antiderivatives. Areas. The content for this lecture is in sections 4.9 and 5.1 Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Antiderivatives A function F is called an antiderivative of f on an interval I if F 0 = f for all x in I . Example: f ( x ) = 2 x Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Theorem: If F is an antiderivative of f then the most general antiderivative of f is F ( x ) + C where C is an arbitrary constant General Antiderivative = Particular Antiderivative + Constant Ryerson University MTH 140, Winter 2012
How to find particular antiderivatives? Ryerson University MTH 140, Winter 2012
How to find particular antiderivatives? Ryerson University MTH 140, Winter 2012
How to find particular antiderivatives? Ryerson University MTH 140, Winter 2012
How to find particular antiderivatives? Ryerson University MTH 140, Winter 2012
How to find particular antiderivatives? Ryerson University MTH 140, Winter 2012
Ryerson University MTH 140, Winter 2012
Two important properties: If F ( x ) is an antiderivative of f ( x ) and G ( x ) is an antiderivative of g ( x ) then cF ( x ) is an antiderivative of cf ( x ) F ( x ) + G ( x ) is an antiderivative of f ( x ) + g ( x ) Ryerson University MTH 140, Winter 2012
Two important properties: If F ( x ) is an antiderivative of f ( x ) and G ( x ) is an antiderivative of g ( x ) then cF ( x ) is an antiderivative of cf ( x ) F ( x ) + G ( x ) is an antiderivative of f ( x ) + g ( x ) This is consequence of the following analysis: ( cF ) 0 = cF 0 = cf ( F + G ) 0 = F 0 + G 0 = f + g Ryerson University MTH 140, Winter 2012
Example: