MATH 149, Spring 2011 Recitation 5 Problems1. Since raindrops grow as they fall, their surface area increases and therefore the resistance to theirfalling increases. A raindrop has initial downward velocity of 10m/s and its downward accelerationisa D(9  0:9t if 0  t  100 if t > 10If 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 3t2 2t C 1. At time t D 1,itisatthepoint 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. Whatis the acceleration?4. Let f and g be functions such that f000.x/ D g000.x/ for all x, f00.0/ D g00.0/, f0.0/ D g0.0/, andf.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 isv D KrLCCCLwhere K and C are knownpositive constants. What is the length of the wave that gives the minimumvelocity?6. Locate the point P D .x; y/ on the quarter-circlex2C y2D 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 with sides 2x, 2, and 2.Find the area A and the x that maximizes it. Hint:IndA=dx simplifyp1  x2 x2=p1  x2to.1  2x2/=p1  x2.8. How large a rectangle fits into the triangle with sides x D 0, y D 0, and x=4 Cy=6 D 1? Find thepoint on this third side that maximizes the area xy.9. Prove thatRbacf.x/ dx D cRbaf.x/ dx using the definition of the definite integral.10. Prove that if f.x/ 0 for a  x  b, thenRbaf.x/ dx 0 using the definition of the definiteintegral.1MATH 149, Spring 2011 Recitation 5 Problems11. Assume that f is continuous on Œ1; 7 and thatZ51f.x/ dx D 3;Z32f.x/ dx D 1;Z53f.x/ dx D 1;Z73f.x/ dx D 6:Evaluate:(a)Z31f.x/ dx (b)Z52f.x/ dx (c)Z72f.x/ dx(d)Z75f.x/ dx (e)Z21f.x/ dx (f)Z71f.x/ dx12. Assume that g is continuous on Œ1; 9 and thatZ41g.x/ dx D 1;Z42g.r/ dr D 2;Z62g.s/ ds D 0;Z96g.t / dt D 1:Evaluate:(a)Z21g.u/ du (b)Z92g.v/ d v (c)Z64g.w/ dw(d)Z91g.x/ dx (e)Z94g.y/ dy (f)Z61g.z/ dz4 April,