Warmup: find the area under the curve from x = 0 to x = 4
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Transcript of Warmup: find the area under the curve from x = 0 to x = 4
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WARMUP: FIND THE AREA UNDER THE CURVE
FROM X = 0 TO X = 4
USE A COUPLE OF DIFFERENT RAMS AND LET’S SEE IF THE ‘ONE TRUE ANSWER’ DOESN’T EMERGE.
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‘AREA’ AS AN EMERGENT PROPERTY (LRAM SIDE)N (# of columns)
Type: L or RRAM?
Result N (# of columns)
Type: L or RRAM?
Result
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‘AREA’ AS AN EMERGENT PROPERTY (RRAM SIDE)N (# of columns)
Type: L or RRAM?
Result N (# of columns)
Type: L or RRAM?
Result
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IN WORDS, THE INTEGRAL (CALCULATION OF AREA) IS THE AREA THAT EMERGES AS WE LET THE N ∞.THE VALUE, THE AREA, ‘EMERGES’ REGARDLESS OF
THE CHOICE OF MODEL!We can approximate this using our calculator by
• Calculating and storing Δx Remember:
• setting up the summation equation, Remember the equation changes for different RAMs.
• Increasing n and re-storing Δx
• Recalling and re-evaluating the summation equation.
If we are comfortable with the calculator, we can generate 5 estimates in a single minute, showing a powerful trend toward a SINGLE value for AREA.
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NEW TASK: THE VELOCITY OF A MODEL PLANE IS GIVEN BY V(T) = 2LN(T + 1) + 8, WITH T IN SECONDS AND VELOCITY IN METERS/SECOND.
Use the process we outlined: through a series of estimates, make a prediction for the “area under the curve” in the first 10 seconds.
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IN CASE WE DON’T TALK ABOUT DURING CLASS:
1. Quiz on calculator prowess Monday2. Also on the table work we do next.3. The “area” from that last slide is really DISTANCE4. It comes with units: meters
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I DRIVE A HONDA CIVIC SI. ROAD AND TRACK ANALYZED THIS CAR A FEW YEARS AGO AND FOUND THESE CHARACTERISTICS:
time Speed0 0
0.6 101.5 202.4 303.7 405 50
6.8 608.7 7011.2 8013.9 9017 100na 110
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HOPEFULLY, I REMEMBERED TO ASK YOU WHAT THE UNITS WERE, WHAT WAS GOING ON, ETC. LET’S USE THIS PAGE TO KEEP TRACK OF THOSE ANSWERS
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BY THE TIME THE CAR IS MOVING AT 100MPH, 17 SECONDS HAS ELAPSED. HOW FAR AWAY IS THE CAR?
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BACK TO THE DATA TABLE: ENTER THE DATA INTO YOUR LISTS; LET’S CONVERT TO FEET / SECOND
(THERE ARE 5,280 FEET IN A MILE AND 60X60 SECONDS IN AN HOUR – LET’S USE A LITTLE DIMENSIONAL ANALYSIS)
time Speed0 0
0.6 101.5 202.4 303.7 405 50
6.8 608.7 7011.2 8013.9 9017 100na 110
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TAKE L2 X 5280 / 3600 AND STORE IT IN …L2
YOUR NEW VALUES COMPARE SECONDS FROM START TO FEET PER SECOND
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AT THIS POINT, IT GET HARD TO RUN CLASS FROM A POWERPOINT….BUT I STILL WANT TO KNOW: HOW MANY “FEET” HAS THE CAR MOVED?
timeseconds
speed in ft / sec
0 0.0000.6 14.6671.5 29.3332.4 44.0003.7 58.6675 73.333
6.8 88.0008.7 102.66711.2 117.33313.9 132.00017 146.667na 161.333