Estimating with Finite Sums

1. Estimating with Finite Sums

2. 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

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

4. 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

5. 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

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

7. A More Realistic Scenario A 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?

8. 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

9. 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 15 Speed (ft/sec) 22 37 41 50 58 63 How far has the train traveled?

10. Example: This is called a LRAM • Graphically: Left –hand endpoint Rectangular Approximation Method Is it an over or under-estimate? Why? Would LRAM ever be an over-estimate?

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

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

13. Example: Time (sec) 0 5 10 15 Speed (ft/sec) 60 50 35 30 A train’s speeds are measured, yielding the data below: • Compute LRAM and RRAM using 3 rectangles LRAM: RRAM:

14. 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

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

16. 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]

17. 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

18. 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