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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

• 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

Time4

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

Time5

Skills for Success• Many factors impact a train’s speed

– Train weight– Train length– Engineer– Track’s curvature– Speed limit– Physical conditions

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A More Realistic ScenarioA train’s speed is measured every 5 seconds, resulting in the following data:

Time (sec) 0 5 10 15 20

Speed (ft/sec) 10 16.25 23 30.25 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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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) 0 3 6 9 12 15Speed (ft/sec) 22 37 41 50 58 63

How far has the train traveled?

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Example:•Graphically: This is called a

LRAMLeft –hand endpoint

Rectangular Approximation Method

Is it an over or under-estimate? Why?

Would LRAM ever be an over-estimate?

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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

LRAM:

RRAM:

Time (sec) 0 5 10 15Speed (ft/sec) 60 50 35 30

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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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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?

• As a train dispatcher, what roles can you have?

• 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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