RIEMANNS SUM Section 4 2 Calculus APDual Revised

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RIEMANN’S SUM Section 4. 2 Calculus AP/Dual, Revised © 2018 viet. dang@humbleisd. net 12/19/2021

RIEMANN’S SUM Section 4. 2 Calculus AP/Dual, Revised © 2018 viet. dang@humbleisd. net 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 1

HISTORY OF BERNHARD RIEMANN 1826 -1866 German Mathematician and math professor Big on Number

HISTORY OF BERNHARD RIEMANN 1826 -1866 German Mathematician and math professor Big on Number Theory and geometry Discovered an estimation of area under the curve through geometry • As an unformal way of estimating area under the curve • • 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 2

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? 12/19/2021 10: 19 AM §

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 3

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Left Handed Rule It is

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Left Handed Rule It is an overestimation 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 4

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Right Handed Sum It is

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Right Handed Sum It is an underestimation 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 5

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Midpoint Rule 12/19/2021 10: 19

HOW DO WE APPROXIMATE THE AREA UNDER THE CURVE? Midpoint Rule 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 6

BREAKDOWN OF RIEMANN’S SUM A. Left side approximation is the left of the sub-interval

BREAKDOWN OF RIEMANN’S SUM A. Left side approximation is the left of the sub-interval is used to determine the height of the rectangle. B. Right side approximation is the right of the sub-interval is used to determine the height of the rectangle. C. Midpoint approximation is the midpoint of each sub-interval us used to determine the height of each rectangle. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 7

EXAMPLE 1 Given the graph below, is the graph of right Riemann’s Sum an

EXAMPLE 1 Given the graph below, is the graph of right Riemann’s Sum an overestimation or underestimation? Explain the answer. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 8

EXAMPLE 2 Given the graph below, is the graph of left Riemann’s Sum an

EXAMPLE 2 Given the graph below, is the graph of left Riemann’s Sum an overestimation or underestimation? Explain the answer. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 9

EXAMPLE 3 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 10

EXAMPLE 3 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 10

YOUR TURN Given the graph below, is the graph of right Riemann’s Sum an

YOUR TURN Given the graph below, is the graph of right Riemann’s Sum an overestimation or underestimation? Explain the answer. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 11

RIEMANN’S SUM 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 12

RIEMANN’S SUM 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 12

File. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 13

File. 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 13

VISUAL 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 14

VISUAL 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 14

STEPS 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 15

STEPS 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 15

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 16

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 16

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 17

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 17

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 18

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 18

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 19

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 19

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 20

EXAMPLE 4 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 20

EXAMPLE 5 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 21

EXAMPLE 5 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 21

EXAMPLE 5 A 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 22

EXAMPLE 5 A 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 22

EXAMPLE 5 B 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 23

EXAMPLE 5 B 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 23

EXAMPLE 5 C 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 24

EXAMPLE 5 C 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 24

EXAMPLE 5 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 25

EXAMPLE 5 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 25

EXAMPLE 6 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 26

EXAMPLE 6 12/19/2021 10: 19 AM § 4. 2: Riemann's Sum 26

EXAMPLE 6 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 27

EXAMPLE 6 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 27

EXAMPLE 6 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 28

EXAMPLE 6 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 28

YOUR TURN 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 29

YOUR TURN 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 29

EXAMPLE 7 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 30

EXAMPLE 7 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 30

YOUR TURN 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 31

YOUR TURN 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 31

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) 12/19/2021 10: 20 AM § 4. 2:

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 32

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Vocabulary 12/19/2021 10: 20 AM Process and

AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) Vocabulary 12/19/2021 10: 20 AM Process and Connections § 4. 2: Riemann's Sum Answer and Justifications 33

ASSIGNMENT Worksheet 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 34

ASSIGNMENT Worksheet 12/19/2021 10: 20 AM § 4. 2: Riemann's Sum 34

POLAR COORDINATES Section 11 5 Calculus APDual RevisedPOLAR COORDINATES Section 11 5 Calculus APDual Revised
INVERSE FUNCTIONS Section 5 3 Calculus APDual RevisedINVERSE FUNCTIONS Section 5 3 Calculus APDual Revised
RELATED RATES Section 2 6 Calculus APDual RevisedRELATED RATES Section 2 6 Calculus APDual Revised
OPTIMIZATION Section 3 7 A Calculus APDual RevisedOPTIMIZATION Section 3 7 A Calculus APDual Revised
PARTICLE MOTION Section 3 7 Calculus APDual RevisedPARTICLE MOTION Section 3 7 Calculus APDual Revised
CONCAVITY Section 3 4 Calculus APDual Revised 2017CONCAVITY Section 3 4 Calculus APDual Revised 2017
INVERSE FUNCTIONS Section 5 3 Calculus APDual RevisedINVERSE FUNCTIONS Section 5 3 Calculus APDual Revised
MOTION Section 7 5 A Calculus APDual RevisedMOTION Section 7 5 A Calculus APDual Revised
SLOPE FIELDS Section 6 1 Calculus APDual RevisedSLOPE FIELDS Section 6 1 Calculus APDual Revised
USUBSTITUTION Section 4 5 Calculus BC APDual RevisedUSUBSTITUTION Section 4 5 Calculus BC APDual Revised
CURVE SKETCHING Section 3 6 Calculus APDual RevisedCURVE SKETCHING Section 3 6 Calculus APDual Revised
Section 9 3 Calculus BC APDual Revised 2018Section 9 3 Calculus BC APDual Revised 2018
ONESIDED LIMITS Section 1 4 Calculus APDual RevisedONESIDED LIMITS Section 1 4 Calculus APDual Revised
4003 Riemanns Sums AP Calculus A Accumulation Rate4003 Riemanns Sums AP Calculus A Accumulation Rate
GROWTH AND DECAY Section 6 2 Calculus APDualGROWTH AND DECAY Section 6 2 Calculus APDual
LAGRANGES ERROR Section 9 7 Calculus BC APDualLAGRANGES ERROR Section 9 7 Calculus BC APDual
NATURAL BASE INTEGRATION Section 5 4 Calculus APDualNATURAL BASE INTEGRATION Section 5 4 Calculus APDual
NATURAL BASE DERIVATIVES Section 5 4 Calculus APDualNATURAL BASE DERIVATIVES Section 5 4 Calculus APDual
SEPARATION OF VARIABLES Section 6 3 Calculus APDualSEPARATION OF VARIABLES Section 6 3 Calculus APDual
SECOND DERIVATIVE TEST Section 3 4 Calculus APDualSECOND DERIVATIVE TEST Section 3 4 Calculus APDual
POWER RULE Section 2 2 Calculus AB APDualPOWER RULE Section 2 2 Calculus AB APDual
POLAR AREA Section 10 5 Calculus BC APDualPOLAR AREA Section 10 5 Calculus BC APDual
MEAN VALUE THEOREM Section 3 2 Calculus APDualMEAN VALUE THEOREM Section 3 2 Calculus APDual
ELLIPSES Section 10 3 Precalculus Pre APDual RevisedELLIPSES Section 10 3 Precalculus Pre APDual Revised