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Unformatted text preview: Calcul diff erentiel et int egral pour les sciences de la vie I MAT1730 Test 1 Professeur: Benoit Dionne Question 1 Find the derivative of the following function : f ( x ) = x 3 ln( x ) 3 x + 7 e x . 4 points Solution: We have f ( x ) = g ( x ) /h ( x ), where g ( x ) = x 3 ln( x ) and h ( x ) = 3 x + 7 e x . Since g ( x ) = 3 x 2 ln( x ) + x 3 parenleftbigg 1 x parenrightbigg = 3 x 2 ln( x ) + x 2 = x 2 (3 ln( x ) + 1) , we have f ( x ) = g ( x ) h ( x ) g ( x ) h ( x ) h 2 ( x ) = x 2 (3 ln( x ) + 1)(3 x + 7 e x ) x 3 ln( x )(3 + 7 e x ) (3 x + 7 e x ) 2 Question 2 Over the course of a year, the city of Ottawa has its highest average monthly temperature 4 points of 21 C in August and its lowest monthly average of 9 C in February. Assume that temperature varies sinusoidally over a period of one year. Find the parameters in the standard cosine description , i.e., f ( x ) = M + A cos parenleftbigg 2 P ( t T ) parenrightbigg , where t is in months, and t = 0 correspond to the month of January. Draw the graph of the function and identify the four parameters A, B, , T in the graph. Give the names of the four parameters. Solution: The mean is M = 21 9 2 = 6, the amplitude is A = 21 + 9 2 = 15, the period is 12 months and the phase is T = 7 months. We get the following graph.= 7 months....
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