MATH 149 Spring 2009 Recitation 5 Problems Wednesday 8 April 2009 1 Since raindrops grow as they fall their surface area increases and therefore the resistance to their falling increases A raindrop has initial downward velocity of 10m s and its downward acceleration is 9 0 9t if 0 t 10 aD 0 if t 0 If the raindrop is initially 500 m above ground how long does it take to fall 2 A particle P moves on a line so that its velocity at time t is 3t 2 2t C 1 At time t D 1 it is at the point 2 Where is P at time t D 3 3 A particle at rest is subjected to a constant acceleration which moves it 60 feet in 4 seconds What is the acceleration 4 Let f and g be functions such that f 000 x D g 000 x for all x f 00 0 D g 00 0 f 0 0 D g 0 0 and f 0 D g 0 Show that f x D g x for all x 5 The velocity of a wave of length L in deep water is r L C vDK C C L where K and C are known positive constants What is the length of the wave that gives the minimum velocity 6 Locate the point P D x y on the quarter circle x2 C y 2 D 1 x 0 y 0 so that the sum of the distances from P to the y axis and to the point 1 0 shall be a maximum What is the maximum 7 Draw a kite that has a triangle with sides 1 1 and 2x next to a triangle p 2 and 2 p with sides 2x 2 2 2 Find the area p A and the x that maximizes it Hint In dA dx simplify 1 x x 1 x to 2 1 2x 1 x 2 8 How large a rectangle fits into the triangle with sides x D 0 y D 0 and x 4 C y 6 D 1 Find the point on this third side that maximizes the area xy 9 Prove that Rb a cf x dx D c Rb a f x dx using the definition of the definite integral 10 Prove that if f x 0 for a x b then integral Rb a 1 f x dx 0 using the definition of the definite MATH 149 Spring 2009 Recitation 5 Problems Wednesday 8 April 2009 11 Assume that f is continuous on 1 7 and that Z Z 5 1 5 3 Evaluate Z 3 f x dx a Z 7 d 5 Z f x dx f x dx D 3 3 2 Z f x dx D 1 2 2 e 1 f x dx D 1 7 f x dx D 6 3 Z 5 b 1 Z Z f x dx c f x dx f 7 2 Z 7 1 f x dx f x dx 12 Assume that g is continuous on 1 9 and that Z 4 1 Z 2 Evaluate Z 2 g u du a Z Z 1 6 6 Z 9 g r dr D 2 g t dt D 1 6 g w dw c 4 Z 9 g y dy e 2 g s ds D 0 g v dv Z 9 4 Z 2 g x dx d g x dx D 1 9 b 1 Z 6 g z dz f 4 1 AMS LATEX 2
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