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.

‘AREA’ AS AN EMERGENT PROPERTY (LRAM SIDE)N (# of columns)

Type: L or RRAM?

Result N (# of columns)

Type: L or RRAM?

Result

‘AREA’ AS AN EMERGENT PROPERTY (RRAM SIDE)N (# of columns)

Type: L or RRAM?

Result N (# of columns)

Type: L or RRAM?

Result

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.

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.

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

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

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

BY THE TIME THE CAR IS MOVING AT 100MPH, 17 SECONDS HAS ELAPSED. HOW FAR AWAY IS THE CAR?

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

TAKE L2 X 5280 / 3600 AND STORE IT IN …L2

YOUR NEW VALUES COMPARE SECONDS FROM START TO FEET PER SECOND

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