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S>0;t>0 Example Suppose the number of miles a car can run before its battery wears out follows the exponential distribution with mean = miles If the owner of the car takes a 5000mile trip what is the probability that he will be able to complete the trip without having to replace the battery of the car?. If g(x) = Z x 0 1 1tt2 dt, find where g is concave up To find where g is concave up, we must compute its second derivative and find where it is positive g0(x) = 1 1xx2 = (1xx2)−1 g00(x) = −(12x) (1xx2)2 Since the denominator is always positive, we need look only for where the numerator is positive. « x z \ w ÿ ú w w Ú ò w F t , n V ^ æ O Ô ù x z p ^ w º 0 { Ø t G å b w q b { ¢ È » Ä w Ð T £ H Ú C « x z !.

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In mathematics, function composition is an operation that takes two functions f and g and produces a function h such that h(x) = g(f(x))In this operation, the function g is applied to the result of applying the function f to xThat is, the functions f X → Y and g Y → Z are composed to yield a function that maps x in X to g(f(x)) in Z. T @ ~ } g X C t Z X z f B O X Ђ̉ߋ ̋ l R ~ ݒn { 茧 s s 钬 P V P ɗL l ͌Â Ȃ 폜 ܂. Z L A E Z b V m O ɁA T o ́A w 肳 ꂽ Ԃ̂ݗL ł ؖ N C A g ɒ񎦂 K v ܂ B ~ ꂽ ڑ ł́A z X g 񎦂 ؖ ͉ߓ ɗL ł A ݂͗L ł͂ ܂ B.

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