Unformatted text preview: Practice Exam for the ﬁrst Midterm (Time: 90 minutes)
1. Evaluate the following integrals: (a) (b)
π 8 x3 +
0 1 x dx e2x + 3 cos (2x) dx (Remark: your ﬁnal answer may not contain any sin or cos-function.) 2. The power which a power station is providing during a 24 hour period to a city is πt given by the function f (t) = 200 + 100 sin 224 . (The time is measured in hours after midnight, the power is measured in megawatts. Power is the rate of change in the energy produced by the power plant.) Answer the following questions: (a) For the the term 100 sin
2πt 24 , what is the amplitude, the period, and the frequency? (b) When is the energy output of the plant greatest and when lowest? (c) What is the average power output between midnight and noon? (d) What is the average power output during the whole day? 3. If F (x) = x3 − 1 and F (1) = 2 what is F (x)? 4. A valve in a full 5000 liter water tank is slowly opening. Water slowly ﬂows out of the tank through a valve at a rate (in liters per hour) of f (t) = 15t2 where t is measured in hours. How long does it take to empty the tank? 5. What is the area of the region between the graphs of the function f (x) = 5x and g (x) = x3 bounded on the left by the y -axis and on the right by the vertical line x = 1? 6. The acceleration of a car after t seconds is a(t) = 2t +1 (measured in at t = 2 was 7 (measured in m ). Answer the following questions: s (a) What is the velocity at t = 4? (b) When is the velocity 31 m ? s (c) What distance has the car travelled between t = 2 and t = 4?
m ). s2 The velocity ...
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- Spring '08
- Derivative, Litre, following questions, Midnight, average power output