Discuss how you would find the area under this curve!
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Transcript of Discuss how you would find the area under this curve!
Discuss how you would find the area under this curve!
AP Calculus Unit 5 Day 2
APROXIMATE Area under a Curve
• We will introduce some new Calculus concepts–LRAM–RRAM–MRAM
Rectangular Approximation Method (RAM) to estimate area under curve
Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].
NOTICE—This is an underestimate. What if the function had been a decreasing function?
2( )f x x
Rectangular Approximation Method (RAM) to estimate area under curve SOLUTION
Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].
NOTICE—This is an underestimate. What if the function had been a decreasing function? over
2( )f x x
(0)(1)+(1)(1)+(4)(1)=5
Is this an overestimate or an underestimate? What if the function had been a decreasing function?
Is this an overestimate or an underestimate? What if the function had been a decreasing function? Under
SOLUTION
(1)(1)+(4)(1)+(9)(1)=14
BEWARE—MRAM uses the average of the two x-values NOT the two y-values!
MRAM gives the best estimate regardless of whether the function is increasing or decreasing.
Midpoint Method SolutionFind the area between the x-axis and y = x2 between [0,3] with partition widths of 1.
MRAM: The Midpoint of each rectangle touches the graph.
x 1.5
f 1.5 2 2 21 0.5 1 1.5 1 2.5 8.75
The middle x-values in each region
CAUTION!!!!MRAM ≠ (LRAM + RRAM)/2
Complete the following table:Increasing Function
Decreasing Function
LRAM
RRAM
Under/Over Estimates for Increasing Function
LRAM produces an _____________underestimate RRAM produces an _____________overestimate
LRAM produces an _____________overestimate RRAM produces an _____________underestimate
Under/Over Estimates for Decreasing Function
How do we get the best underestimate of the area between a and b?
a b
Answer: Divide into 2 intervals (increasing/decreasing) and pick LRAM or RRAM for each part that yields an underestimate for that half.
Together let’s practice with finding partition widths
Find the upper and lower approximations of the area under
on the interval [1,2] using 5 partitions. What is the width of each partition?
( ) 2f x x Find the upper and lower approximations of the area under on the interval [0,2] using 8 partitions. What is the width of each partition?
1( )f xx
Area Under Curves
How could we make these approximations more accurate?
You Try:
3 0 3 3 3 6 12.545
3 3 3 6 3 9 21.545
3 1.5 3 4.5 3 7.5 18.254
LRAM
RRAM
MRAM
Let’s look at a HW problem . . .
Unit 5 HW Document
Problem #5
LRAM - Using a table
x 0 3 6 9 12f(x) 20 30 25 40 37
LRAM 3 20
Partitions
Partition [0,3] [3,6] [6,9] [9,12]
3 30 3 25 3 40 345
f(x) value at left end of interval.
Width of interval
RRAM - Using a table
x 0 3 6 9 12f(x) 20 30 25 40 37
RRAM 3 30
Partitions
Partition [0,3] [3,6] [6,9] [9,12]
3 25 3 40 3 37 396
f(x) value at right end of interval.
Width of interval
Midpoint Rectangle Sums (MRAM)
x 0 2 4 6 8 10 12f(x) 20 30 25 40 42 32 37
Partitions
Partition midpoints
MRAM 4 30 4 40 4 32 408
Use the middle function value in each interval.
Ex: Calculate the Midpoint Sum using 3, equal-width partitions.
NOTES: Background Problem
• A car is traveling at a constant rate of 55 miles per hour from 2pm to 5pm. How far did the car travel?– Numerical Solution—
– Graphical Solution—
• But of course a car does not usually travel at a constant rate . . . .
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Example Problem:
Time (secon
ds
0 2 4 6 8 10
Speed (ft/sec
)
30 36 40 48 54 60
“never decreasing”—
Scatterplot
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Example Problem:
1. What is the best lower estimate for the distance the car traveled in the first 2 seconds?
“lower estimate”--_________ since increasing function
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Example Problem:
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
2. What is the best upper estimate for the distance the car traveled in the first 2 seconds?
You Try: 3. What is the best lower estimate for the total distance traveled during the first 4 seconds? (Assume 2 second intervals since data is in 2 second intervals)
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
You Try: 4. What is the best upper estimate for the total distance traveled during the first 4 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
You Try: 5. Continuing this process, what is the best lower estimate for the total distance traveled in the first 10 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
You Try: 6. What is the best upper estimate for the total distance traveled in the first 10 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
Riemann SumsThese estimates are sums of products and are known as Riemann Sums. 7. If you choose the lower estimate as your approximation of how far the car traveled, what is the maximum amount your approximation could differ from the actual distance?