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105-3-AveragesPartsTrigIntegrals

105-3-AveragesPartsTrigIntegrals - 3 Odd and Even Functions...

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3 Odd and Even Functions, Averages, Integration by Parts & Trigonometric Integrals 3.1 Odd and Even Functions Definition A function f ( x ) is called: Even if f ( - x ) = f ( x ) for all x in the domain of f ; Odd if f ( - x ) = - f ( x ) for all x in the domain of f . Some examples: f ( x ) = x 2 is even because f ( - x ) = ( - x ) 2 = x 2 = f ( x ) , f ( x ) = x 3 is odd because f ( - x ) = ( - x ) 3 = - x 3 = - f ( x ) , x 6 - 1 - x 2 is even because ( - x ) 6 - p 1 - ( - x ) 2 = x 6 - 1 - x 2 , cos( x ) is even because cos( - x ) = cos( x ) , sin( x ) is odd because sin( - x ) = - sin( x ) , x sin( x ) is even because ( - x ) sin( - x ) = x sin( x ) , tan( x ) is odd because tan( - x ) = sin( - x ) cos( - x ) = - sin( x ) cos( x ) = - tan( x ) , Note: most function are neither odd nor even. Symmetry These two properties have clear meanings in terms of symmetry of the graph of the function: If f ( x ) is even, then its graph y = f ( x ) is symmetric in the y -axis. This means that if you reflect it in the y -axis, in other words you look at the graph y = f ( - x ), then that’s the same graph as y = f ( x ). If f ( x ) is odd, then its graph y = f ( x ) is symmetric in the origin. This means that if you reflect it in the y -axis and then in the x -axis, and get the graph y = - f ( - x ), then that’s the same graph as y = f ( x ). Integrals of Odd and Even Functions If you know that a function is odd or even, then that can help you do some integrals of it, namely the ones that have symmetric limits like R a - a . For even functions, the left part of the graph will have the same area as the right part, so we only have to compute one and multiply it by two. For odd functions, the left part of the graph will have the same area as the right part, but counted negatively, so the areas of the left and right part will cancel each other out, and the integral equals 0. If f ( x ) is odd and continuous on [ - a, a ], then Z a - a f ( x ) dx = 0. If f ( x ) is even and continuous on [ - a, a ], then Z a - a f ( x ) dx = 2 Z a 0 f ( x ) dx . The second one will merely make the calculation a bit more efficient (plugging in 0 is usually easy), but the first one can make the entire calculation redundant. 1
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A few examples of integrals of odd and even functions (check for yourself if the function is odd or even): Z 7 - 7 x 4 dx = 2 Z 7 0 x 4 dx = 2 · 1 5 x 5 7 0 = 2 5 · (7 5 - 0) = 2 5 · 7 5 , Z 1 - 1 tan( x ) dx = 0 , Z 2 - 2 x 5 cos( x ) dx = 0 . 3.2 Averages and the Mean Value Theorem Averages You already know what a discrete average is: for instance, if f ( k ) gives your score on the k -th homework, and there are n homeworks in total, then your average score is f = 1 n n X k =1 f ( k ) . This is called a discrete average because this is a discrete function, which means it only takes values on isolated numbers. In calculus however, we deal with functions that take values on all numbers in an interval [ a, b ], so we couldn’t take a sum like above. What we can do instead is take the integral over the interval, and we divide by the length of the interval (similarly to dividing by n above).
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