124quiz1solutions

# 124quiz1solutions - Solution We model the function H t as a...

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Math 124 Quiz 1 Sept. 1, 2006 SOLUTIONS 1. Solve the following equation for x : 2 e 3 x = 4 e 5 x Solution: Take a natural log of both sides: ln(2 e 3 x ) = ln(4 e 5 x ) ln 2 + 3 x = ln 4 + 5 x ln 2 - ln 4 = 2 x x = ln(1 / 2) 2 2. The population of a region is growing exponentially. There were 40 million people in 1980 ( t = 0) and 56 million in 1990. (a) Find an expression for the population at any time t , in years. (Hint: start with the general form of an exponential function). (b) What population would you predict for the year 2000? (c) What is the doubling time? Solution: (a) Start with the general exponential function P = P 0 a t . We let t = 0 correspond to 1980, and hence P 0 = 40. The remaining task is to ﬁnd a . We plug in 10 for t and 56 for P : 56 = 40 a 10 56 40 = a 10 a = ± 7 5 ² 1 10 Hence, P = 40 ± 7 5 ² 1 10 t (b) P (20) = 40 ( 7 5 ) 20 10 = 40 ± 7 5 ² 2 (c) We solve 80 = 40 ( 7 5 ) 1 10 t . Hence, 2 = ( 7 5 ) t 10 . ln 2 = t 10 ln ± 7 5 ² t = 10 ln 2 ln ( 7 5 ) 1

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3. The desert temperature, H , oscillates daily between 70 degrees Farenheit at 5am and 100 degrees at 5pm. Write a possible formula for H in terms of t , measured in hours from 5am.
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Unformatted text preview: Solution: We model the function H ( t ) as a sinusoidal function. The graph below shows temperature ( H ) as a function of the time t in hours. This is a phase-shifted cosine function with amplitude 15 and period 24. There is also a vertical shift of 85 units up. Since the period = 2 π B , we see that B = π 12 . H ( t ) = 85 + 15 cos ± π 12 ( t-12) ² 4. Draw a graph of the function f ( x ) = arctan x . Label all important behavior. 2 5. Find a possible formula for the function g ( x ) whose graph is given below. Solution: Notice that there are three roots (zeros) of g ( x ): -3, 0, and 4. Hence, g ( x ) = k ( x + 3) x ( x-4) for some constant k . We can determine k by using the fact that (2 , 3) is on the graph: 3 = g (2) = k (2 + 3)(2)(2-4) 3 =-20 k k = 3-20 Hence, g ( x ) =-3 20 ( x + 3) x ( x-4) 3...
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124quiz1solutions - Solution We model the function H t as a...

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