test 1C soln - Calcul direntiel et intgral pour les...

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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 5 ln( x ) 4 x + 6 e x . 4 points Solution: We have f ( x ) = g ( x ) /h ( x ), where g ( x ) = x 5 ln( x ) and h ( x ) = 4 x + 6 e x . Since g ( x ) = 5 x 4 ln( x ) + x 5 parenleftbigg 1 x parenrightbigg = 5 x 4 ln( x ) + x 4 = x 4 (5 ln( x ) + 1) , we have f ( x ) = g ( x ) h ( x ) g ( x ) h ( x ) h 2 ( x ) = x 4 (5 ln( x ) + 1)(4 x + 6 e x ) x 5 ln( x )(4 + 6 e x ) (4 x + 6 e x ) 2 Question 2 Over the course of a year, the city of Ottawa has its highest average monthly temperature of 4 points 23 C in August and its lowest monthly average of 3 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 = 23 3 2 = 10, the amplitude is A = 23 + 3 2 = 13, the period is 12 months and the phase is T = 7 months. We get the following graph.
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2 4 6 8 10 4 8 8 4 y y = f ( t ) t 12 12 16 20 24 28 10 13 7 2 12 14
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