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Unformatted text preview: 6.1, #9. The graph is that given by figure 6.5 on page 309 with one change: the graph y = 50 for the truck starts at the yaxis. The two graphs intersect twice, at t = 0.7 and t = 4.3. At the beginning the truck is ahead of the car. The first intersection gives the time when the lead of the truck is maximal. The car then overtakes the truck sometime between 1 pm and 2 pm. The second intersection gives the time when the lead of the car is maximal.
3 6.2, #1.
1 b a (x2  x) dx = 1 3 3 1 3x2 dx  1 2 3 2x dx =
1 26 8 14  = . 3 2 3 6.2, #2. 1 dx is the area of the rectangle with top on the line y = 1, bottom on the b xaxis, and sides on the lines x = a and x = b. So a 1 dx = b  a. In particular,
5 8 3 3 a dx = 3,
2 3 1 dx = 11,
1 23 dx = 23
1 1 dx = 46. 6.2, #3. a x dx for 0 a b is the area of the trapezoid with top on the line y = x, bottom on the xaxis, and sides on the lines x = a and x = b. So
b b x dx =
a 1 1 (b + a)(b  a) = (b2  a2 ). 2 2
8 8 3 The formula holds when a 0 b. For example, 3 x dx = 3 x dx since 3 x dx = 0 by the symmetry of graph of y = x with respect to the origin. b a The formula also holds when a b 0. For a x dx =  b x dx by the geometric interpretation of the integrals. In particular,
5 8 3 x dx = 21/2,
2 3 x dx = 55/2,
1 5x dx = 5 4 = 20. 6.2 #8. The definite integral 0 ex dx can be interpreted as the area between the graphs 2 of y = 0 and y = ex between the vertical lines x = 0 and x = 1. This area is positive 2 2 since 0 < ex for 0 < x 1. Since e < 3 and the function f (x) = ex is increasing on [0, 2 1] it follows that ex < 3 for 0 x 1. Comparing areas we see that
1 0 1 2 ex dx <
0 2 1 3dx = 3(1)  3(0) = 3. ...
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This note was uploaded on 07/06/2011 for the course MATH 180 taught by Professor Tan during the Fall '08 term at Ill. Chicago.
 Fall '08
 TAN
 Calculus

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