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Congratulations!
• You are now a train dispatcher
• Train Dispatchers are the air traffic controllers
of the railroads. They control the movement of
trains over large track territories
• They use computers and radio communications
to control the safe movement of trains
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Job Requirements
•
•
•
•
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Communication skills
Math
Science
Attentive to detail
Safety conscious
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Skills for Success
• Understanding speed, time and distance
• If a train travels at a constant speed of 50 ft/sec
for 20 seconds, what distance does it travel?
Speed
Time
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Skills for Success
• How do you determine the distance graphically, if the train is
now traveling at a constant speed of 35 ft/sec for 20 seconds,
from 1:00:00 p.m. to 1:00:20 p.m.?
• After 20 seconds you check the speed again and determine the
train is now traveling at 40 ft/sec and continues at this speed
for 20 seconds. Determine the overall distance.
• What is the difference between the rectangles?
Speed
Time
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Skills for Success
• Many factors impact a train’s speed
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Train weight
Train length
Engineer
Track’s curvature
Speed limit
Physical conditions
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A More Realistic Scenario
A train’s speed is measured every 5 seconds, resulting in the
following data:
Time (sec)
Speed (ft/sec)
0
10
5
16.25
10
23
15
30.25
20
38
Can you approximate how far the train traveled?
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Exploring Three Methods
• Rectangular Approximation Methods
– Left-hand endpoint (LRAM)
– Right-hand endpoint (RRAM)
– Midpoint (MRAM)
• The height of the rectangle is determined by
the method used
• Approximations
– Over-estimate
– Under-estimate
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Summary
• All approximations compute using Δx·∑f(xi)
• LRAM: Dx[f(0) + f(1) + f(2) + … + f(n-1)]
• MRAM: Dx[f(M1) + f(M2) + … + f(MN)]
• RRAM: Dx[f(1) + f(2) + f(3) + … + f(n)]
• Δx = B-A
N
• M is the midpoint
• N = the number of intervals
• A and B are the range
Example:
• How do we measure distance traveled?
A train is moving with increasing speed. We measure the
train’s speed every three seconds and obtain the following
data.
Time (sec)
Speed (ft/sec)
0 3 6 9 12
22 37 41 50 58
15
63
How far has the train traveled?
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Example:
• Graphically:
This is called a
LRAM
Left –hand endpoint
Rectangular Approximation
Method
Is it an over or under-estimate?
Why?
Would LRAM ever be an overestimate?
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Example:
• Graphically:
This is called a
RRAM
Right –hand endpoint
Rectangular Approximation
Method
Is it an under or over-estimate?
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Example:
• This time let’s take the midpoint:
This is called an
MRAM
Midpoint Rectangular
Approximation Method
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Example:
A train’s speeds are measured, yielding the data below:
• Compute LRAM and RRAM using 3 rectangles
Time (sec)
0
Speed (ft/sec) 60
5
50
10
35
15
30
LRAM:
RRAM:
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Over or Under Estimates
• If f(x) is decreasing
– LRAM is an over-estimate
– RRAM is an under-estimate
– Rn < area < Ln
• If f(x) is increasing
– LRAM is an under-estimate
– RRAM is an over-estimate
– Ln < area < Rn
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Example:
What if we changed the number of intervals?
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RAM
• If you increase the number of intervals (rectangles), your
approximations become increasingly more accurate
• What if we take the limit as the number of intervals→∞?
– This should give us the exact area under the curve
• If f(x) is continuous on [a,b], then the endpoint and
midpoint approximations approach one and the same
limit L:
– lim RN = lim LN = lim MN = L
N→∞
N→∞
N→∞
• If f(x)>0 on [a,b], we take L as the definition of the area
under the graph of y = f(x) over [a,b]
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Example:
• Sketch the graph of y = 8 – x between the
region and the x-axis 1<x<5 into 4 subintervals.
• Compute LRAM, MRAM and RRAM.
n=
a=
b=
LRAM: Dx[f(0) + f(1) + f(2) + … + f(n-1)]
RRAM: Dx[f(1) + f(2) + f(3) + … + f(n)]
MRAM: Dx[f(m1) + f(m2) + f(m3) +… + f(mn)]
Example:
• Sketch the graph of y = 2 + 4x – x2 between the
region and the x-axis 0<x<4 into 6 subintervals.
• Compute LRAM, MRAM and RRAM.
LRAM: Dx[f(0) + f(1) + f(2) + … + f(n-1)]
RRAM: Dx[f(1) + f(2) + f(3) + … + f(n)]
MRAM: Dx[f(m1) + f(m2) + f(m3) +… + f(mn)]
Summary
• We can approximate the area under a curve using rectangles
• There are three methods for approximating:
– LRAM
– MRAM
– RRAM
The method will determine the heights of the approximating rectangles
• Increasing the number of rectangles makes the approximation more accurate
• The increasing/decreasing nature of the curve impacts the accuracy of the
estimate
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Summary
• Why is finding an area important?
• It is a way to describe how the instantaneous
changes accumulate over an interval
• We call this integral calculus
• Other applications
– The work it takes to empty a tank of oil
– Accumulation of water in a conical tank
– The pressure against a dam at any depth
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