hard questions?

Question

http://www.webassign.net/scalcet/5-1-001alt.gif

a = 10
(a) By reading values from the given graph of f, use five
rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x = 0 to x = 20.

? lower estimate
? upper estimate

(b) Find new estimates using ten rectangles in each case.

? lower estimate
? upper estimate


The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance, d that she traveled during these three seconds.
t(s) 0 0.5 1.0 1.5 2.0 2.5 3.0
v(ft/s) 0 6.7 10.8 15.5 18.8 19.4 20

d=(lower estimate)
d=(upper estimate)

Answer 1

a)
x 0, 4, 8, 12, 16, 20
f(x) 2, 6, 9, 10.5, 12.5, 14 (with variation)
delta x = 20/5= 4
Lower sum 4[2+ 6+ 9+ 10.5+ 12.5]= 160
Upper sum 4[6+ 9+ 10.5+ 12.5+ 14]= 208
b)
x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
f(x) 2, 4.5, 6, 7.5, 9, 9.5, 10.5, 11.5, 12.5, 13, 14 (with variation)
delta x = 20/10= 2
Lower sum 2[2+ 4.5+ 6+ 7.5+ 9+ 9.5+ 10.5+ 11.5+ 12.5+ 13]= 172
Upper sum 2[ 4.5+ 6+ 7.5+ 9+ 9.5+ 10.5+ 11.5+ 12.5+ 13+ 14]= 196
c) Integrate (estimate area) velocity for distance
t(s) 0 0.5 1.0 1.5 2.0 2.5 3.0
v(ft/s) 0 6.7 10.8 15.5 18.8 19.4 20
delta x = 3/6 =1/2
Lower sum 1/2[0+ 6.7+ 10.8+ 15.5+ 18.8+ 19.4]= 35.6
Upper sum 1/2[ 6.7+ 10.8+ 15.5+ 18.8+ 19.4+ 20]= 45.6

Answer 2

For the 5 rectangles from x = 0 to x = 20, split the domain into 5 equal pieces: [0,4], [4,8], ...[16,20] (Since the diagram is only 12 squares wide, it looks like each interval should be 2 squares wide.)

Each of these has width 4 and a height which varies from one end of the interval to the other. If you use the lowest height (in this case the one for the lower end of the interval) then you will get an underestimate of the true area of the vertical strip; if you use the highest height (the one for the higher end of the interval) you'll get an overestimate for the area of the rectangle. Either way, compute the area of each rectangle and sum to get the lower and upper estimates for the total area under the graph.

Next repeat the same procedure with 10 rectangles. Each will have a width of 2 instead of 4 (i.e. be one square wide instead of two). Since each interval is smaller, the difference between the height at the beginning and the height at the end of each interval will be smaller, so the two estimates will be closer together and more accurate than the estimates with only 5 rectangles.

For the runner problem, distance = speed X time. If you graph speed against time, the area is the distance. The procedure is the same except each interval should be half a second wide (and since you have the values, you don't actually have to look at a graph.)

Answer 3

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