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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 4 ln( x ) 2 x + 8 e x . 4 points Solution: We have f ( x ) = g ( x ) /h ( x ), where g ( x ) = x 4 ln( x ) and h ( x ) = 2 x + 8 e x . Since g ′ ( x ) = 4 x 3 ln( x ) + x 4 parenleftbigg 1 x parenrightbigg = 4 x 3 ln( x ) + x 3 = x 3 (4 ln( x ) + 1) , we have f ′ ( x ) = g ′ ( x ) h ( x ) − g ( x ) h ′ ( x ) h 2 ( x ) = x 3 (4 ln( x ) + 1)(2 x + 8 e x ) − x 4 ln( x )(2 + 8 e x ) (2 x + 8 e x ) 2 Question 2 Over the course of a year, the city of Ottawa has its highest average monthly temperature 4 points of 26 ◦ C in August and its lowest monthly average of − 8 ◦ 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 π 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, Φ , T in the graph. Give the names of the four parameters. Solution: The mean is M = 26 − 8 2 = 9, the amplitude is A = 26 + 8 2 = 17, the period is 12 months and the phase is T = 7 months. We get the following graph. 2 2 4...
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