mt2_2009_key

# mt2_2009_key - Key for Math 102-106 - November 2009 Midterm...

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Key for Math 102-106 - November 2009 Midterm 2 1. Find the maximum of the function V ( r ) = r 2 e - r where r is a positive number. Using V 0 ( r ) ﬁnd the critical point V 0 ( r ) = 2 re - r - r 2 e - r = e - r (2 r - r 2 ) . Thus, V 0 ( r ) = 0 when r = 0 and r = 2 . Use second derivative to check the type of critical points V 00 ( r ) = 2 e - r - 4 re - r + r 2 e - r = e - r (2 - 4 r + r 2 ) , so V 00 (0) = 2 > 0 (and r = 0 is a minimum) and V 00 (2) = - 2 e - 2 < 0 . Therefore, when r = 2 , V ( r ) reaches its maximum value of V (2) = 4 e - 2 . 2. Two spherical balloons are connected so that one inﬂates as the other deﬂates, the sum of their volume remains constant. When the ﬁrst balloon has radius 20 cm and its radius is increasing at 5 cm/s. the second balloon has radius 10 cm. What is the rate of change of the radius of the second balloon at this time? (Recall that the volume of a sphere of radius r is V = 4 3 πr 3 ). Let

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## This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at UBC.

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mt2_2009_key - Key for Math 102-106 - November 2009 Midterm...

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