Chapter 06 · Guided self-study

Intro + external course study

Calculus: change and accumulation

Connect position, velocity, and acceleration, then integrate a distributed quantity.

2 core ideas2 source trails3 recall prompts

Change map

Local change accumulatesA slope describes what happens now; an area collects what happened across an interval.

Mathematics · Modules

Build the idea

An original introduction, worked example and practice, followed by a curated reading sequence. The external courses supply deeper teaching and problem sets.

Scope and safety

This guide is an orientation, not a complete university module. Practical work needs suitable facilities, safety review and supervision.

Understand the idea

Start with these prerequisites

Algebra for physical problems Geometry, graphs, and trigonometry Motion, forces, and energy

Shared science foundations

Derivatives and local change ↗Integrals and accumulation ↗

Calculus connects local change with accumulated effect. A derivative describes change near an instant; an integral accumulates contributions over an interval.

Module lens

Change and accumulation connect

Derivatives describe local change while integrals collect a quantity over an interval.

Core relationshipv = dx/dt · Δx = ∫v dt

Track: position · velocity · time
Ask yourself: What does the slope mean in this graph?

Original Lilerno schematic · conceptual, not to scale

Differentiate a physical function

If position is x(t), its derivative is velocity and the derivative of velocity is acceleration. Constants in a physical equation carry units; x = 2t² only describes metres when the coefficient has units m/s².

Integrate with limits and a reference

Integrating velocity from one time to another gives displacement, not necessarily distance travelled. An indefinite integral needs an integration constant. Initial conditions tell you which member of a family of functions describes the system.

Change map

Local change accumulatesA slope describes what happens now; an area collects what happened across an interval.

See the relationship

Change one quantity, watch the graph respond, then explain the result in your own words.

Predict → change → explain

From a secant to a tangent

Shrink Δt. Which value does the secant slope approach?

1Position 2t²
2Two points
3Δx / Δt
4Limit: velocity 4t

Time t (s)1Interval Δt (s)1

Secant slope: 6 m/s
Derivative: 4 m/s

Position
Tangent
Secant

Axes: Time (s) → Position (m). Bounds may rescale when inputs change.

Secant = 6 m/s; tangent = 4 m/s. Their difference is 2Δt.

Model assumptions and limits

x(t) = 2t² metres, with time in seconds. The derivative is 4t m/s; the secant spans t to t + Δt.

Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.

Inspect the plotted values

Time (s) and Position (m); values rounded to four significant figures.
Series	Time (s)	Position (m)
Position	0	0
Position	0.05	0.005
Position	0.1	0.02
Position	0.15	0.045
Position	0.2	0.08
Position	0.25	0.125
Position	0.3	0.18
Position	0.35	0.245
Position	0.4	0.32
Position	0.45	0.405
Position	0.5	0.5
Position	0.55	0.605
Position	0.6	0.72
Position	0.65	0.845
Position	0.7	0.98
Position	0.75	1.125
Position	0.8	1.28
Position	0.85	1.445
Position	0.9	1.62
Position	0.95	1.805
Position	1	2
Position	1.05	2.205
Position	1.1	2.42
Position	1.15	2.645
Position	1.2	2.88
Position	1.25	3.125
Position	1.3	3.38
Position	1.35	3.645
Position	1.4	3.92
Position	1.45	4.205
Position	1.5	4.5
Position	1.55	4.805
Position	1.6	5.12
Position	1.65	5.445
Position	1.7	5.78
Position	1.75	6.125
Position	1.8	6.48
Position	1.85	6.845
Position	1.9	7.22
Position	1.95	7.605
Position	2	8
Position	2.05	8.405
Position	2.1	8.82
Position	2.15	9.245
Position	2.2	9.68
Position	2.25	10.13
Position	2.3	10.58
Position	2.35	11.05
Position	2.4	11.52
Position	2.45	12.01
Position	2.5	12.5
Position	2.55	13
Position	2.6	13.52
Position	2.65	14.04
Position	2.7	14.58
Position	2.75	15.13
Position	2.8	15.68
Position	2.85	16.25
Position	2.9	16.82
Position	2.95	17.41
Position	3	18
Position	3.05	18.6
Position	3.1	19.22
Position	3.15	19.84
Position	3.2	20.48
Position	3.25	21.13
Position	3.3	21.78
Position	3.35	22.45
Position	3.4	23.12
Position	3.45	23.81
Position	3.5	24.5
Position	3.55	25.2
Position	3.6	25.92
Position	3.65	26.64
Position	3.7	27.38
Position	3.75	28.13
Position	3.8	28.88
Position	3.85	29.65
Position	3.9	30.42
Position	3.95	31.21
Position	4	32
Tangent	0	-2
Tangent	0.025	-1.9
Tangent	0.05	-1.8
Tangent	0.075	-1.7
Tangent	0.1	-1.6
Tangent	0.125	-1.5
Tangent	0.15	-1.4
Tangent	0.175	-1.3
Tangent	0.2	-1.2
Tangent	0.225	-1.1
Tangent	0.25	-1
Tangent	0.275	-0.9
Tangent	0.3	-0.8
Tangent	0.325	-0.7
Tangent	0.35	-0.6
Tangent	0.375	-0.5
Tangent	0.4	-0.4
Tangent	0.425	-0.3
Tangent	0.45	-0.2
Tangent	0.475	-0.1
Tangent	0.5	0
Tangent	0.525	0.1
Tangent	0.55	0.2
Tangent	0.575	0.3
Tangent	0.6	0.4
Tangent	0.625	0.5
Tangent	0.65	0.6
Tangent	0.675	0.7
Tangent	0.7	0.8
Tangent	0.725	0.9
Tangent	0.75	1
Tangent	0.775	1.1
Tangent	0.8	1.2
Tangent	0.825	1.3
Tangent	0.85	1.4
Tangent	0.875	1.5
Tangent	0.9	1.6
Tangent	0.925	1.7
Tangent	0.95	1.8
Tangent	0.975	1.9
Tangent	1	2
Tangent	1.025	2.1
Tangent	1.05	2.2
Tangent	1.075	2.3
Tangent	1.1	2.4
Tangent	1.125	2.5
Tangent	1.15	2.6
Tangent	1.175	2.7
Tangent	1.2	2.8
Tangent	1.225	2.9
Tangent	1.25	3
Tangent	1.275	3.1
Tangent	1.3	3.2
Tangent	1.325	3.3
Tangent	1.35	3.4
Tangent	1.375	3.5
Tangent	1.4	3.6
Tangent	1.425	3.7
Tangent	1.45	3.8
Tangent	1.475	3.9
Tangent	1.5	4
Tangent	1.525	4.1
Tangent	1.55	4.2
Tangent	1.575	4.3
Tangent	1.6	4.4
Tangent	1.625	4.5
Tangent	1.65	4.6
Tangent	1.675	4.7
Tangent	1.7	4.8
Tangent	1.725	4.9
Tangent	1.75	5
Tangent	1.775	5.1
Tangent	1.8	5.2
Tangent	1.825	5.3
Tangent	1.85	5.4
Tangent	1.875	5.5
Tangent	1.9	5.6
Tangent	1.925	5.7
Tangent	1.95	5.8
Tangent	1.975	5.9
Tangent	2	6
Secant	1	2
Secant	2	8

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Open learning, traceable sources

Read further

Checked 2026-09-06

Original Lilerno lessons and diagrams, supported by these references. Free access does not always permit republication. Links open the publisher’s material.

18.02SC · Multivariable Calculus (2010)MIT OpenCourseWare

Continue after single-variable calculus: partial derivatives, double and triple integrals, vector fields and flux.

CC BY-NC-SA 4.0 except separately credited material. Linked for external study, not reproduced. Some assigned textbooks/software require separate access.

Open source ↗Rights / publisher record ↗

18.01SC · Single Variable CalculusMIT OpenCourseWare · Fall 2010

Derivatives, integrals, and worked calculus study sessions.

CC BY-NC-SA, except marked exceptions; external study only.

Open source ↗Rights / publisher record ↗

Understand the idea

Start with these prerequisites

Algebra for physical problems Geometry, graphs, and trigonometry Motion, forces, and energy

Shared science foundations

Derivatives and local change ↗Integrals and accumulation ↗

Calculus connects local change with accumulated effect. A derivative describes change near an instant; an integral accumulates contributions over an interval.

Module lens

Change and accumulation connect

Derivatives describe local change while integrals collect a quantity over an interval.

Core relationshipv = dx/dt · Δx = ∫v dt

Track: position · velocity · time
Ask yourself: What does the slope mean in this graph?

Original Lilerno schematic · conceptual, not to scale

Differentiate a physical function

Integrate with limits and a reference

Change map

Local change accumulatesA slope describes what happens now; an area collects what happened across an interval.

See the relationship

Change one quantity, watch the graph respond, then explain the result in your own words.

Predict → change → explain

From a secant to a tangent

Shrink Δt. Which value does the secant slope approach?

1Position 2t²
2Two points
3Δx / Δt
4Limit: velocity 4t

Time t (s)1Interval Δt (s)1

Secant slope: 6 m/s
Derivative: 4 m/s

Position
Tangent
Secant

Axes: Time (s) → Position (m). Bounds may rescale when inputs change.

Secant = 6 m/s; tangent = 4 m/s. Their difference is 2Δt.

Model assumptions and limits

x(t) = 2t² metres, with time in seconds. The derivative is 4t m/s; the secant spans t to t + Δt.

Original Lilerno illustration. Inputs are illustrative; this is not experimental evidence or a design rating.

Inspect the plotted values

Time (s) and Position (m); values rounded to four significant figures.
Series	Time (s)	Position (m)
Position	0	0
Position	0.05	0.005
Position	0.1	0.02
Position	0.15	0.045
Position	0.2	0.08
Position	0.25	0.125
Position	0.3	0.18
Position	0.35	0.245
Position	0.4	0.32
Position	0.45	0.405
Position	0.5	0.5
Position	0.55	0.605
Position	0.6	0.72
Position	0.65	0.845
Position	0.7	0.98
Position	0.75	1.125
Position	0.8	1.28
Position	0.85	1.445
Position	0.9	1.62
Position	0.95	1.805
Position	1	2
Position	1.05	2.205
Position	1.1	2.42
Position	1.15	2.645
Position	1.2	2.88
Position	1.25	3.125
Position	1.3	3.38
Position	1.35	3.645
Position	1.4	3.92
Position	1.45	4.205
Position	1.5	4.5
Position	1.55	4.805
Position	1.6	5.12
Position	1.65	5.445
Position	1.7	5.78
Position	1.75	6.125
Position	1.8	6.48
Position	1.85	6.845
Position	1.9	7.22
Position	1.95	7.605
Position	2	8
Position	2.05	8.405
Position	2.1	8.82
Position	2.15	9.245
Position	2.2	9.68
Position	2.25	10.13
Position	2.3	10.58
Position	2.35	11.05
Position	2.4	11.52
Position	2.45	12.01
Position	2.5	12.5
Position	2.55	13
Position	2.6	13.52
Position	2.65	14.04
Position	2.7	14.58
Position	2.75	15.13
Position	2.8	15.68
Position	2.85	16.25
Position	2.9	16.82
Position	2.95	17.41
Position	3	18
Position	3.05	18.6
Position	3.1	19.22
Position	3.15	19.84
Position	3.2	20.48
Position	3.25	21.13
Position	3.3	21.78
Position	3.35	22.45
Position	3.4	23.12
Position	3.45	23.81
Position	3.5	24.5
Position	3.55	25.2
Position	3.6	25.92
Position	3.65	26.64
Position	3.7	27.38
Position	3.75	28.13
Position	3.8	28.88
Position	3.85	29.65
Position	3.9	30.42
Position	3.95	31.21
Position	4	32
Tangent	0	-2
Tangent	0.025	-1.9
Tangent	0.05	-1.8
Tangent	0.075	-1.7
Tangent	0.1	-1.6
Tangent	0.125	-1.5
Tangent	0.15	-1.4
Tangent	0.175	-1.3
Tangent	0.2	-1.2
Tangent	0.225	-1.1
Tangent	0.25	-1
Tangent	0.275	-0.9
Tangent	0.3	-0.8
Tangent	0.325	-0.7
Tangent	0.35	-0.6
Tangent	0.375	-0.5
Tangent	0.4	-0.4
Tangent	0.425	-0.3
Tangent	0.45	-0.2
Tangent	0.475	-0.1
Tangent	0.5	0
Tangent	0.525	0.1
Tangent	0.55	0.2
Tangent	0.575	0.3
Tangent	0.6	0.4
Tangent	0.625	0.5
Tangent	0.65	0.6
Tangent	0.675	0.7
Tangent	0.7	0.8
Tangent	0.725	0.9
Tangent	0.75	1
Tangent	0.775	1.1
Tangent	0.8	1.2
Tangent	0.825	1.3
Tangent	0.85	1.4
Tangent	0.875	1.5
Tangent	0.9	1.6
Tangent	0.925	1.7
Tangent	0.95	1.8
Tangent	0.975	1.9
Tangent	1	2
Tangent	1.025	2.1
Tangent	1.05	2.2
Tangent	1.075	2.3
Tangent	1.1	2.4
Tangent	1.125	2.5
Tangent	1.15	2.6
Tangent	1.175	2.7
Tangent	1.2	2.8
Tangent	1.225	2.9
Tangent	1.25	3
Tangent	1.275	3.1
Tangent	1.3	3.2
Tangent	1.325	3.3
Tangent	1.35	3.4
Tangent	1.375	3.5
Tangent	1.4	3.6
Tangent	1.425	3.7
Tangent	1.45	3.8
Tangent	1.475	3.9
Tangent	1.5	4
Tangent	1.525	4.1
Tangent	1.55	4.2
Tangent	1.575	4.3
Tangent	1.6	4.4
Tangent	1.625	4.5
Tangent	1.65	4.6
Tangent	1.675	4.7
Tangent	1.7	4.8
Tangent	1.725	4.9
Tangent	1.75	5
Tangent	1.775	5.1
Tangent	1.8	5.2
Tangent	1.825	5.3
Tangent	1.85	5.4
Tangent	1.875	5.5
Tangent	1.9	5.6
Tangent	1.925	5.7
Tangent	1.95	5.8
Tangent	1.975	5.9
Tangent	2	6
Secant	1	2
Secant	2	8

Your progress0 reviews

Ready to move on?

Save your place when you finish reading.

Self-reported recall, not an assessment of mastery.

Loading progress…

Calculus: change and accumulation

Understand the idea

Change and accumulation connect

Differentiate a physical function

Integrate with limits and a reference

From a secant to a tangent

Ready to move on?

18.01SC · Single Variable Calculus ↗

18.02SC · Multivariable Calculus (2010) ↗

Your study record

Read further

Calculus: change and accumulation

Understand the idea

Change and accumulation connect

Differentiate a physical function

Integrate with limits and a reference

From a secant to a tangent

Ready to move on?

18.01SC · Single Variable Calculus ↗

18.02SC · Multivariable Calculus (2010) ↗

Your study record

Read further