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Unformatted text preview: Math 151, Workshop 13 1. Suppose F (x) = 1 f (t) dt where f is the function whose graph is displayed. The graph consists of three line segments (for x between 0 and 3) followed by an unknown curve. Also, the value of 5 f (t)dt is - 2 . 3 0 a) What is F (5)? b) Find the equation of the line tangent to the graph of F (x) at the point (3, F (3)) .
-1 -2 -3 3x2 + 6x + 2 dx = ln 4. 1 x(x + 1)(x + 2) Hint: Begin by differentiating the function ln(x + 2) + ln(x + 1) + ln x.
2 x 2 1 y t
0 1 2 3 4 5 2. Verify that 3. A waste holding tank for an industrial process is constructed as shown to the right. The cross-sectional area of the holding tank, a cylinder, is 5 square feet, and the tank's height is 15 feet. Assume that the effluent is entering the top of the tank using the sluiceway shown and the rate of fluid entering the tank is modeled by the periodic function f (t) = 3.5 + sin 2 t . f (t) is measured in cubic feet per hour 24 and t is measured in hours, with t = 0 being midnight. Each of the pipes which empty the tank has a carrying capacity of 2 cubic feet per hour. One pipe is always open. The other pipe is open from t = 12 (noon) until t = 24 (the next midnight). You may assume that when a pipe is open, its carrying capacity is fully used. The tank at time t = 0 contains 10 cubic feet of fluid, so the fluid depth is 2 feet. What is the depth at the next midnight, when t = 24? Does the fluid overflow the tank during the first 24 hours, where 0 t 24? If this model is accurate, does the fluid ever overflow the tank? IN OUT 4. Suppose that f is a continuous function (defined for all x) and that the values of the following integrals are known:
1 0 f (x) dx = 5 ; 1 -1 f (x) dx = 3 ; 2 0 f (x) dx = 8 ;
3 2 4 0 f (x) dx = 11 . Evaluate the following integrals: a)
2 0 f (2x) dx b) 0 sin xf (cos x) dx c) xf (8 - x2 ) dx. ...
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- Spring '08