x dx 2 20pts A function f x is odd if f x f x and even if f x f x a Explain why

# X dx 2 20pts a function f x is odd if f x f x and

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- x ) dx 2. (20pts) A function f ( x ) is odd if f ( - x ) = - f ( x ) and even if f ( - x ) = f ( x ). (a) Explain why 3. (15pts) A chain suspended from the points ( - 1 , 0) and (1 , 0) has the shape of a cate- nary, given by the formula f ( x ) = cosh( x ) - cosh(1). What is the length of the chain? 4. (20pts) A thin plate P of uniform density 1 fills the area below the curve y = x (1 - x ) and above the x -axis. Sketch the area and find the ( x, y ) coordinates of the center of mass of P. 5. (15pts) The improper integral integraldisplay 0 e - x 2 2 dx = radicalbigg π 2 , so p ( x ) = radicalbigg 2 π e - x 2 2 defines a proba- bility density function on the domain x 0. (a) Find the mean μ of a random variable with probability density function p ( x ). (b) (Challenge) The variance σ 2 of this random variable is defined as σ 2 = integraldisplay 0 ( x - μ ) 2 p ( x ) dx. Evaluate this improper integral. If you are unsure about your numeric answer for μ , you can leave μ in your answer as an unknown constant.

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• Spring '13
• BruceFontaine
• Math, Calculus, Variance, Probability theory, probability density function, density function