q2-sol-1042-sp08

q2-sol-1042-sp08 - Calculus II--1042 4 Solution Quiz 2 1....

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Unformatted text preview: Calculus II--1042 4 Solution Quiz 2 1. Write the sum 4 k=1 (-1)k cos (-1)k cos k , without sigma notation and then evaluate it. k = (-1)1 cos() + (-1)2 cos( ) + (-1)3 cos + (-1)4 cos 2 3 4 2 1 = (-1) (-1) + (1) (0) + (-1) ( ) + (1) 2 2 1+ 2 = 1 + 0 - 1/2 + 2/2 = 2 Solution: k=1 2. Given table shows the velocity of a model train engine moving along a track for 10 minutes. Estimate the distance traveled by the engine using subintervals of equal time length 2 minutes with Mid-points Rule. Time(Min) Velocity(ft/min) 0 0 1 40 2 75 3 90 4 110 5 105 6 80 7 60 8 35 9 20 10 0 Solution: Width h = 2 minutes is given. So the partition and their mid-points are Time(min.) Mid-points Velocity(ft/min) 0 1 40 2 3 90 4 5 105 6 7 60 8 9 20 10 Distance= 40 2 + 90 2 + 105 2 + 60 2 + 20 2 = 80 + 180 + 210 + 120 + 40 = 690 ft. 3. If f (x) = 9 - x2 , on [0, 4]. Parition the interval into FOUR subintervals of equal length. Compute the Riemann's Sum using (a) Left-end Points. (Sketch the graph and shaded the associated triangles) Solution: a = 0, b = 4, and n = 4 9 Width of rectangle is h = x x0 x1 x2 x3 x4 0 1 2 3 4 Estimated area using left end points is then M4 = f (0) 1 + f (1) 1 + f (2) 1 + f (3) 1 = 9 1 + 8 1 + 5 1 + 0 1 = 9 + 8 + 5 + 0 = 13 Squared units 4 4-0 = = 1. 4 4 7 Using h = 1 the partition is y=9-x^2 5 3 1 -1 0 -2 -5 1 2 3 4 (b) Mid-Points Rule. (Sketch the graph and shaded the associated triangles) Solution: a = 0, b = 4, and n = 4 9 4 4-0 = = 1. Using h = 1 the partition is 4 4 x x0 x1 x2 x3 x4 0 1 2 3 4 Mid-points 1/2 3/2 5/2 7/2 Estimated area using Mid points is then Width of rectangle is h = 7 5 3 1 -1 0 -2 -5 1 2 y=9-x^2 3 4 M4 = f (1/2) 1 + f (3/2) 1 + f (5/2) 1 + f (7/2) 1 1 9 25 49 = 9- + 9- + 9- + 9- 1 4 4 4 4 35 27 11 -13 35 + 27 + 11 - 13 60 = + + + = = = 15 Squared units 4 4 4 4 4 4 ...
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This homework help was uploaded on 04/18/2008 for the course MATH 1042 taught by Professor Dr.z during the Spring '08 term at Temple.

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