Review from Thursday Use RRAM LRRAM and MRAM

Review from Thursday Use RRAM, LRRAM, and MRAM with 3 partitions to estimate

AB Calculus Unit 5 Day 3 More Riemann Sums and Applications

Area Under Curves You Try: How could we make these approximations more accurate?

Complete the following table: Increasing Function LRAM RRAM Decreasing Function

Under/Over Estimates for Increasing Function LRAM produces an _______ underestimate RRAM produces an _______ overestimate

Under/Over Estimates for Decreasing Function LRAM produces an _______ overestimate RRAM produces an _______ underestimate

LRAM - Using a table Partitions x 0 3 6 9 12 f(x) 20 30 25 40 37 Partition Width of interval [0, 3] [3, 6] [6, 9] f(x) value at left end of interval. [9, 12]

RRAM - Using a table Partitions x 0 3 6 9 12 f(x) 20 30 25 40 37 Partition Width of interval [0, 3] [3, 6] [6, 9] f(x) value at right end of interval. [9, 12]

Midpoint Rectangle Sums (MRAM) Use the middle function value in each interval. Ex: Calculate the Midpoint Sum using 3, equal-width partitions. Partitions x 0 2 4 6 8 10 12 f(x) 20 30 25 40 42 32 37 Partition midpoints

Application of Integrals A BIG application of integrals is they take a RATE of change and find an ACCUMULATION over time.

Riemann Sums are tools for ESTIMATING integrals, area, and total accumulation

NOTES: Background Problem • A car is traveling at a constant rate of 55 miles per hour from 2 pm to 5 pm. How far did the car travel? – Numerical Solution— – Graphical Solution— • But of course a car does not usually travel at a constant rate. .

Example Problem: 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. Time (secon ds 0 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60 “never decreasing”— Scatterplot

Example Problem: 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. Time (sec) 0 2 4 6 8 10 Speed (ft/sec) 30 36 40 48 54 60 1. What is the best lower estimate for the distance the car traveled in the first 2 seconds? “lower estimate”--_____ since increasing function

Example Problem: 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. 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