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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 2

Job Requirements Communication skills Math Science Attentive to detail Safety conscious 3

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 4

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 5

Skills for Success Many factors impact a train’s speed –Train weight –Train length –Engineer –Track’s curvature –Speed limit –Physical conditions 6

A More Realistic Scenario A train’s speed is measured every 5 seconds, resulting in the following data: Time (sec) Speed (ft/sec) Can you approximate how far the train traveled? 7

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 8

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) How far has the train traveled? 9

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 over- estimate? 10

Example: Graphically: This is called a RRAM Right –hand endpoint Rectangular Approximation Method Is it an under or over-estimate? 11

Example: This time let’s take the midpoint: This is called an MRAM Midpoint Rectangular Approximation Method 12

Example: A train’s speeds are measured, yielding the data below: Compute LRAM and RRAM using 3 rectangles LRAM: RRAM: Time (sec) Speed (ft/sec)

Over or Under Estimates If f(x) is decreasing –LRAM is an over-estimate –RRAM is an under-estimate –R n < area < L n If f(x) is increasing –LRAM is an under-estimate –RRAM is an over-estimate –L n < area < R n 14

Example: What if we changed the number of intervals? 15

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 R N = lim L N = lim M N = 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] 16

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 17

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 18