advanced road profile analysis-fundamental calculus

Advanced Road Profile Analysis-

Fundamental Calculus

PURDUE ROAD SCHOOL

INDIANA, 2014

Christopher R. Byrum, Ph.D., P.E. Soil & Materials Engineers, Inc.

Plymouth, MI Office

Profile data was “written” onto a paper roll

N.A.S.-N.R.C Publication 1061, “The AASHO Road Test”, 1962

AASHO Profilometer

Device measures a continuous SLOPE profile

SLOPE VARIANCE found strongly correlated to expert panel

AASHO data analyzed using Electric-Analog Chart Recorder

Test Site Locations

Over 1,800 profiles analyzed

MEPDG SHRP/LTPP Data

GPS3 and GPS4 Sites

HIGH-SPEED PROFILER

Height Sensor

Special Vehicle

inside

Accelerometer

Distance Measurement

Computer

For most LTPP Data: *1 Elevation measurement per inch, raw data.

*Elevation reported every 6” (Avg. of surrounding 12, 1” samples)

* 6” Data also has 300-ft Long Wavelength Reduction Filter

50 mph

PAVEMENT ELEVATION PROFILE:

Uniformly Spaced Point Elevations

IRI Units = L/L Suspension Movement per Mile

Most Common “Roughness Index”

Profile z = f(x)

r2 = 0.93

Std Error = 15.7

Error Plot:

0 50 100 150 200 250 300

Measured Total Faults

GPS3 Fault Model: No Dowels

0 5 10 15 20 25 30 35 40

Pavement Age, years

Measured Faulting

Predicted Faulting

MEPDG IRI and Faulting

Agg Only Dowels

Range Range INPUT DATA Case 1 Case 2 Case 3 Case 4 Case 5

-0.1 to 0.43 -0.09 to +0.32 Slab Curvature, ft-1

x 1000 0.1 0.15 0.2 0.25 0.3

8 to14.3 6.4 to 13.3 PCC thickness, in. 11 11 11 11 11

2 to 6.6 mill. 3.35 to 5.68 PCC E-modulus, psi 4700000 4700000 4700000 4700000 4700000

460 to1000 466 to 738 PCC Split Tensile, psi 600 600 600 600 600

11 to 20 14 to 30 Joint Spacing, ft 20 20 20 20 20

100 to 680 103 to 376 Subgrade Overburden, psf 220 220 220 220 220

2.5 to 34 4.5 to 27.5 Subgrade w% 12 12 12 12 12

2 to 92 1 to 97 Subgrade P200% 25 25 25 25 25

33 to 181 40 to 170 Days with precip/yr 151 151 151 151 151

0 to 3396 0 to 2280 Freeze Index, degF-days/yr 1211 1211 1211 1211 1211

6 to 58 8 to 57 Ann. Precipitation, in 38 38 38 38 38

1 to 203 2 to 187 Days < 32 degF/yr 164 164 164 164 164

0 to 106 3 to 176 Days > 90 degF/yr 5 5 5 5 5

18 to 1470 18 to 820 Initial KESAL/yr 300 300 300 300 300

-0.05 to 0.2 -0.05 to 0.17 Traffic Growth Rate, %/yr 0.05 0.05 0.05 0.05 0.05

45000 Total Faulting at 20-yr 60 72 87 104 122

Avg. Fault at 20-yr 2.4 2.9 3.5 4.2 4.9

Avg KESAL/yr (15-yr) 425 425 425 425 425

seed value, mm/500 feet 56.54 56.54 56.54 56.54 56.54

delay, yr -9.7 -8.1 -7.7 -7.6 -7.6

growth rate 0.02 0.03 0.03 0.04 0.04

MaxFlt - Cum Trf parameter ######## ######## ######## ######## ########

0 5 10 15 20 25 30 35 40Pavement Age, yr

ft Case 1

Case 2

Case 3

Case 4

Case 5

Joints With Dowels

Model Data Range

0 5 10 15 20 25 30 35 40Pavement Age, yr

ft Case 1

Case 2

Case 3

Case 4

Case 5

Aggregate Interlock Only-No Dowels

SPREADSHEET

Same Model Form

Different Coefficients

Traffic Model

Data Ranges

Input Table

Trend Graphs

M EId z

212 1( )

3-D Analyses:

2-D Analyses:

PAVEMENT STRESS ANALYSIS:

Soil-Structure Interaction Engineering

M EId z

212 1( )

3-D Analyses:

2-D Analyses:

PAVEMENT STRESS ANALYSIS:

Soil-Structure Interaction Engineering

SLAB/BEAM

CURVATURE

PAVEMENT PROFILE:

A 2-D Slice through a 3-D Solution LWP

Profile Elevation, z = f(x)

Major Lesson #1 Continuity

Function z = f(x)

z Discontinuities

No Fault Fault

Pavements are “Piecewise Continuous Functions”

with Fuzz (texture, measurement error…)

Function z = f(x)

z Points of Inflection

Function z = f(x)

Downward Curvature

Upward Curvature

Function z = f(x)

SLOPE = “rise/run” =

First Derivative dx

z R, Radius of Curvature 1/ =

CURVATURE of f(x) =

Second Derivative

G P S 3 S i t e 5 5 - 3 0 0 9

5 0 6 0 7 0 8 0 9 0 1 0 0

D i s t a n c e , m

NOTES:

1. Highest IRI of all GPS3 pavements less than 10 years old.

2. High faulting at an early age.

3. Sandy clay subgrade (PI=14.5, w%=13), sand and gravel with 11% P200 base.

4. Built in 1984, 300-350 KESAL/yr, 220 mm PCC, no dowels.

Profile Date: 6/19/1992

ProfileTime: 3:51:44 PM

10± mm

Profile Date: June 19, 1992

Profile Time: 15:51:44

* “Thermally” Curved-Down at this time of day?

* High Level of “Locked-In” Up-Warp and Faulting.

Curved Way Up

Example Profile, z = f(x)

slopeslopecurvature

zzslope

Finite Difference Methods are Ideal

For Constant Interval Profile Data

BREAKING DOWN PROFILE DATA

Discrete Slope

0 15 30 45 60 75Distance, m

Discrete Curvature

0 15 30 45 60 75

Distance, m

Elevation

0 15 30 45 60 75Distance, m

Upward Translation = “Rocking”

Moment Hinge

55-3009

z = f(x)

Discrete Slope

0 15 30 45 60 75Distance, m

Discrete Curvature

0 15 30 45 60 75

Distance, m

Elevation

0 15 30 45 60 75Distance, m

Upward Translation = “Rocking”

Moment Hinge

55-3009

Initial Shape.

Patterns Grow

z = f(x)

Consider Profile as a Matrix

X1 L1 R1

X2 L2 R2

X3 L3 R3

Xn Ln Rn

Raw Profile

Split Profile into 2 Sub-Matrices

X1 L1 R1

X2 L2 R2

X3 L3 R3

Xn Ln Rn

Raw Profile Discontinuities Continuous

Segments

Discontinuities Continuous

Segments

Faulting Evaluation

Discontinuities Continuous

Segments

Slab Curvature, Waves, Tilt, and

Texture Evaluation

Typical Faulting distress

Analysis of Discontinuities

Typical Faulting distress

Fault Size, H

Travel Direction

Structural Discontinuity in

the Road Surface Layer

How much suspension movement

will occur when the IRI quarter

Car goes over the Fault of Size H?

Fault Size, H

Travel Direction

Fault Size, H

Travel Direction

Transient Responses with

Exponential/Linear decay,

A function of damping/friction

about 20 m per cycle

about 2 m per cycle

Fault Size, H

Travel Direction

Current MEPDG Design Guide, IRIH = 1.5 H

Fault Size, H

Travel Direction

Current MEPDG Design Guide , IRIH = 1.5 H

Results of My Study, IRIH = 1.75 H

Site ID: 553008

Profile Date: 7/29/98

ProfileTime: 6:53:01

500-ft Summary Statistics LWP RWP

# Imperfections detected 34 33

Average Size, mm 8.8 11.1

Maximum Size, mm 14.2 16.7

Minimum Size, mm 2.4 4.4

Total Imperfections, mm 298.7 367.8

Total Imperfections, m/km 1.96 2.41

Calculated IRI, m/km 4.42 5.43

The Primary Cause of IRI- Faults

Faulting vs. Roughness for GPS3 Profiles

IRI(faults) = 1.762(TotalFaults) + 1.104

R2 = 0.83

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Total sum of profile faults, m/km

Fault Gain Factor

All GPS3 Data

Y-intercept

WHAT CAN WE UNDERSTAND

FROM FIRST-DERIVATIVE ANALYSIS

Subgrade- Clay, Silt, Sand, Gravel, Muck, Peat…

Subbase Base

JOINTED CONCRETE PAVEMENT

UNDER-SLAB EROSION - pumping

Subbase Base

Trapped Water

Traffic Direction

Sediment Transport

Tilting Data

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

LWP and RWP shown

All-Data

6-in Interval Slope Distribution

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

LWP and RWP shown

All-Data

Faults

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

LWP and RWP shown

All-Data

Wisconsin 19’-18’-13’-14’ Slab Pattern

Tilted

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

LWP and RWP shown

All-Data

Wisconsin 19’-18’-13’-14’ Slab Pattern

Level Tilted

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

All-Data

Slab Slope Data Std. Dev. = f(texture, warp, bumps…)

6-in Interval Slope

553009 19-Jun-92 15:51:44

100120140

160180200

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

0 50 100 150 200 250 300 350 400 450 500

Distance, feet

All-Data

Fault Slope Data

Slab Slope Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

Continuous Slabs-Only

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

NEW PAVEMENT

Slabs-Only

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

INITIATION OF

FAULTING

Slabs-Only

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

FAULTING(t)

Slabs-Only

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

ROCKING(t) FAULTING(t)

Slabs-Only

SLABS ONLY

Without FAULTS

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

ROCKING FAULTING

Slabs-Only

SLABS ONLY

Without FAULTS

All-Data

6-in Interval Slope

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015

Slope Magnitude

Slabs-Only

SLABS ONLY

Without FAULTS

AVERAGE SLAB

SLOPE INDEX

All-Data

6-in Interval Slope

The “mean” slab

Average Fault Spacing, L

RI H = ½(Rocking Fault Size)

Average Per Slab Volume ½(L/2)(H)(Lane Width)

Per Mile Volume Index = (Per Slab Volume)*5280/L

RI = Slab Rocking Index, reported in inches per 10 feet

Rocking Fault Size = RI(L)/10

Rocking

Deformation

ALL DATA LWP RWP

maximum slope 0.0067 0.0078

minimum slope -0.0250 -0.0335

average slope -0.0001 -0.0001

slope stdev 0.0043 0.0052

median slope 0.0007 0.0010

SLABS ONLY LWP RWP

maximum slope 0.0067 0.0068

minimum slope -0.0057 -0.0045

average slope 0.0011 0.0014

slope stdev 0.0022 0.0022

median slope 0.0013 0.0015

slope trimmed mean (remove outer 40%) 0.0012 0.0015

Slab Rocking Index-inches per 10-ft = 0.142 0.182Average Joint/Crack Spacing, feet = 15.2 15.2

Average Fault Size from Rocking Index, mm = 5.48 7.00

Average Fault Size from Profile Data, mm = 5.28 6.85

Volume Pumped, cubic feet per mile per 12-ft lane = 142 182

Average, cubic feet per mile per 12-ft lane = 162

GPS3 Test Group

0 5 10 15 20 25

Pavement Age, yr

0 5 10 15 20 25

Pavement Age, yr

0 5 10 15 20 25

Pavement Age, yr

Generally Constant Rate per ESAL !!!!

For LTPP Site 55-3009

18,900 ESAL per lb-pumped per Slab

WHAT CAN WE UNDERSTAND FROM

SECOND-DERIVATIVE ANALYSIS

Subbase

JOINTED CONCRETE PAVEMENT

CURLING AND WARPING

Subbase Base

Trapped Water

55-3009

z = f(x)

340 345 350 355 360 365

Distance, feet

What is the Curvature of This?

GPS3 55-3009

“Structurally Continuous”

CURLED & WARPED SLAB

Includes: Macro/Mega Texture, Slab Creep, Warp, Curl,

Abrasion, Wheel-Path Location, Sensor Accuracy, Inertial Plane,

Possible Plastic Hinges (pre-hair-line cracking)……….many..

340 345 350 355 360 365

Distance, feet

d 0.0141 feet

Lstringline

STRING-LINE METHOD

•Generally under-estimates

•Wont quantify variability

3-Point Circle/Arc

Curvature = ''2

zd (1)

)0(')(')(''0

Mean Value Theorem for “Continuous”

Slab Functions from Calculus

340 345 350 355 360 365

Distance, feet

Slope = -0.00235

Slope = +0.00372

)0(')(')(''0

340 345 350 355 360 365

Distance, feet

Slope = -0.00235

Slope = +0.00372

)0(')(')(''0

All Have Same Average Curvature

But Different Statistics: std. deviation,

median, max., min…..

SINE-WAVE PERTUBATION

Average Curvature Over These Spans = Zero!

Slope = Zero Slope = Zero

AVG CONCEPT NOT VALID!

•Free Rotation at Hinges, = undefined, infinity or zero

•Vertical Offsets = Faults, = undefined, not connected

• through Fault equal and opposite to Actual Avg Slab

Discontinuity!!! E

levati

340 345 350 355 360 365

Distance, feet

6th Order

2nd Order

)0(')(')(''0

L BEST-FIT POLYNOMIALS

•End slopes vary and can be way off

•Wont quantify variability

z = f(x)

SLOPE at Zero SLOPE at L/2 = 0

L, average slab length

Westergaard’s Infinite Strip Shape Function

z’= dx

zzslope

z’’= 2

slopeslopecurvature

BCI- For Discrete Profile Data

340 345 350 355 360 365

Distance, feet

BCI Method for Single Slabs

End Slope Samples = Average

Std. Deviation = Variation of

34 Curvature Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

End Slope Samples

per Wheel Path

340 345 350 355 360 365

Distance, feet

BCI Method for Single Slab Segments

End Slope Samples

•0.5 to 4 feet from ends

•16 End Slopes per Slab

•8 Curvature Strings per Slab

• 392 Arc Samples

per Wheel Path

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

0.5 Interval Range for Curvature (ft-1

Global

GLOBAL STATISTICS-All Arcs

Standard Deviation = 1.67

#arcs = 8753

Std Error of Mean = 0.018

Avg = 0.498

Median = 0.502

TrimMean = 0.498

4-ft Arc STATISTICS

#arcs = 889

Avg = 0.502

Median = 0.501

TrimMean = 0.511

8-ft Arc STATISTICS

#arcs = 401

Avg = 0.448

Median = 0.458

TrimMean = 0.448

1-ft Arc STATISTICS

#arcs = 1422

Avg = 0.499

Median = 0.656

TrimMean = 0.532

NOTE: 3-5 ft arc length classes are best

= 0.3048 m-1

Arc Class

GPS3 Site 55-3009 has 33 Slabs in 500 feet

340 345 350 355 360 365

Distance, feet

Slope = -0.00235

Slope = +0.00372

up down up

Lets Compare The Methods

-0.001

-0.0005

0.0005

0.0015

0.0025

342 344 346 348 350 352 354 356 358 360 362 364

Distance, feet

2nd Order

3rd Order

4th Order

5th Order

6th Order

4-ft Arcs

3-ft Arcs

5-ft Arcs

Constant

Linear

Parabolic

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

0.5 Interval Range for Curvature (ft-1

Global

GLOBAL STATISTICS-All Arcs

#arcs = 8753

Avg = 0.498

Median = 0.502

TrimMean = 0.498

4-ft Arc STATISTICS

#arcs = 889

Avg = 0.502

Median = 0.501

TrimMean = 0.511

8-ft Arc STATISTICS

#arcs = 401

Avg = 0.448

Median = 0.458

TrimMean = 0.448

1-ft Arc STATISTICS

#arcs = 1422

Avg = 0.499

Median = 0.656

TrimMean = 0.532

NOTE: 3-5 ft arc length classes are best

= 0.3048 m-1

Arc Class

GPS3 Site 55-3009 has 33 Slabs in 500 feet

Avg 0.5/1000 ft-1

= 1/R, R = 2000 ft

For GPS3 Site

55-3009

50% = +/- 0.067

80% = +/- 0.121

90% = +/- 0.166

95% = +/- 0.206

Confidence Intervals For Day-Time Average Slab Curvature

0 10 20 30 40 50 60

Annual Precipitation, inches

0Mean Value

Mean Trend 3 Deg F per inch Gradient

Warp Variation is Huge, down or up

50% = +/- 0.067

80% = +/- 0.121

90% = +/- 0.166

95% = +/- 0.206

Confidence Intervals For Day-Time Average Slab Curvature

0 10 20 30 40 50 60

Annual Precipitation, inches

0Mean Value

Individual

Test Sites

Average of Cluster = Warp

Variation of Cluster = Curl

“Moving Arcs & Slopes”

Continuous

Segments

1. “Moving” 1-ft to 8-ft long arcs are

calculated along each continuous

segment at 0.5 ft intervals

2. Avg., Std. Dev., Median, Count, Std

Error…. for each Arc Length Class

“Generalized Analysis of

Variation of Slope and Curvature

for Discrete Data”

Elevation

350 355 360 365 370 375 380 385 390 395 400Distance, ft.

3 Faults

55-3009 Classic Real Faults with Fine Texture

Tilt Faults

Curvature

IRITotal = IRIFaults + IRIWarp + IRITexture + IRIWaves

For PCC and AC Pavements

The Primary Cause of IRI- Faults

Faulting vs. Roughness for GPS3 Profiles

IRI(faults) = 1.762(TotalFaults) + 1.104

R2 = 0.83

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Total sum of profile faults, m/km

Fault Gain Factor

All GPS3 Data

Y-intercept

Avg. Slab Curvature vs. IRI

Sensitivity Analysis for the IRI Regression Model

Faulting Avg. Slab Curvature Stdev Curvature All Parameters

"Average" LTPP GPS3 Profile

Very Curved

All High

All Low

IRITotal = IRIFaults + IRIWarp + IRITexture + IRIWaves

For PCC and AC Pavements

IRITexture, m/km = 0.065(6 Texture Band Size, mm)

2mm (very fine texture) < 6 < 10 mm (very coarse texture)

0.2 mm Sample Interval

0.5 mm Laser Footprint Diameter

MiDOT Texture Laser Sample Pattern Used

No use having really short spacing with big footprint, low resolution,…..

……………………..

I-69 Tex F 12:34:09 9-Jul-03

ASTM MPD, mm = 1.486

ASTM ETD, mm = 1.389

ASTM Peak Value, mm = 1.721

ASTM Avg Height, mm = 0.235

Texture Tortuosity, mm/mm = 1.161

Macro Median Value, mm = 0.410

Macro Spline StDev, mm = 0.849

Residue StDev, mm = 0.251

# of Sharp 1.1 mm (+) = 22

Avg Size of 1.1+, mm = 2.24

Total Sum of 1.1+, mm = 49.38

Avg Spacing of 1.1+, mm = 8.70

CVT for 1.1 = 2.80

% too smooth = 0.078

4mm "smooth" stdev, mm = 0.20

Curvature 0.2 stdev = 91.49

0.4 stdev = 30.11

0.6 stdev = 16.84

0.8 stdev = 11.16

1.0 stdev = 8.03

1.2 stdev = 6.27

1.4 stdev = 5.06

1.6 stdev = 4.35

SRI(0.3 without), lb/lb = 0.427

SRI(0.3 with), lb/lb = 0.381

%Contact = 68.4%

Tire Bottom Elev, mm = -0.073

Volume Beneath Tire, in^3 = 0.468

Area Beneath Tire, in^2 = 0.067

# Voids beneath rubber = 15

Avg. Void Width, in = 0.14121

Avg Void Depth, in = 0.03156

0.5" Flow Distance Index, mm = 333.4

Avg Shear Stress, psi = 15.17

Median Shear Stress, psi, = 13.80

Shear Stress StDev, psi = 11.30

Max Shear Stress, psi, = 66.35

MacroTexture + Residue (microtexture + scanner "noise")

-3.000

-2.000

-1.000

0 25 50 75 100 125 150 175 200

Distance, mm

Texture Height Distribution

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3Height, mm

quency

1 mm Spline

1 mm Median

+/- 50% line

Spline Residue

Texture Elev. Proportion Curve

0 10 20 30 40 50 60 70 80 90 100

% greater than

Texture Shear Stress

0 5 10 15 20 25 30 35 40 45 50Shear Stress, psi

quency

M EId z

Michigan’s State Fossil:

Mastodon

advanced road profile analysis-fundamental calculus

Documents

4.4 the fundamental theorem of calculus the fundamental

§12.5 the fundamental theorem of calculus

aim: fundamental theorem of calculus course: calculus do...

topic-2-the fundamental theorem of calculus

5.4 fundamental theorem of calculus objectives swbat:...

fundamental theorem of calculus. part i

6034 fundamental theorem of calculus (part 2)

lesson 27: the fundamental theorem of calculus

special focus: fundamental theorem of calculus

lesson 32: the fundamental theorem of calculus

lesson 25: the fundamental theorem of calculus

4.5 the fundamental theorem of calculus

fundamental theorem of wiener calculus

lesson 28: the fundamental theorem of calculus

ap calculus free responses categorized by topic · ap...

calculus preliminaries - clemson...

1.1 the fundamental theorem of calculus

3. fundamental theorem of calculus. fundamental theorem of...

the fundamental theorem of calculus inverse operations

basic integration 8.1 – fundamental theorem ... - calculus