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10FallHwk07_Soln

10FallHwk07_Soln - Math 2413 1(2 pts Homework Set 7...

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Math 2413 Homework Set 7 – Solutions 10 Points 1. (2 pts) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 4 3 2 2 3 3 2 2 2 2 4 4 1 4 16 4 1 4 4 4 1 4 1 4 4 4 4 4 1 4 8 16 g z z z z z z z z z z z z z z z ¢ = - - + + - + Ø ø Ø ø = - + - + + - = - + - - + º û º û The five critical points are then, ( ) 2 1 4 2 1 16 4 0 2 1 4 0 8 16 0 1 513 1.4781, 1.3531 z z z z z z z - = = – + = ° = - + - = = - – = - 4. (2 pts) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 2 1 2 2 2 1 3 3 3 2 2 3 2 2 2 3 6 8 7 2 4 2 8 2 8 7 2 4 2 8 8 7 3 8 7 2 5 40 53 3 8 7 t t t t P t t t t t t t t t t t t t - - - + + - - ¢ = - - + + - - - + = - + - - + = - + Critical points are then where the derivative is zero ( i.e. where the numerator is zero) and where the derivative doesn’t exist ( i.e. where the denominator is zero). For the critical points we need to recall that the function also has to exist at these points (this is especially true for the second set in this case). From the numerator we get the two critical points : ( ) 1 5 20 3 15 1.6762, 6.3238 t = = . From the denominator we get the two critical points : 1, 7 t = (function exists at these two points!). 7. (2 pts) From #1 we know that the critical points are : 1 4 2, , 1.4781, 1.3531 z = – - - . However the only ones that are in the given interval are : 1 4 2, , 1.4781 z = - - - . So evaluating the function at these three critical points and the endpoints of the interval gives, ( ) ( ) ( ) ( ) ( ) 1 4 2 0 0 1.47814 1918.8172 2.1 504.0743 1 5625 g g g g g - = - = - = - = =
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