A

Guide To Calculus

Baseline Rendering

Baseline vs. Progressive Rendering

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Circle and Sphere Fun Facts

3

3

4r24 r2rr2

Dissecting a Circle

~Disc Rings

Unroll the Rings

r2

r 2)2)((2

1rrr

Area = ½ base x height

X-Ray

Original Rings Slices Boards

Ring By Ring

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Slice By Slice

Board By Board

Strategy Visualization Step-by-Step Layout Single Step Zoom

Ring-by-ring

Slice-by-slice

Board-by-board

r

p

x

dp

dx

dr

Strategy Visualization Step-by-Step Layout Calculus Lingo

Ring-by-ring

Slice-by-slice

Board-by-board

Circle Rings Slices Boards

Sphere Shells Wedges Plates

Ring-by-ringTimelapse

Shell-by-ShellX-Ray

Plate-by-plateTimelapse

Circumference Area Volume Surface area

Derivative Symbol Mindset

Think about the resulting step-by-step pattern

Think about the actual ratio of changes

x

1x

1d Perimeter Change 2d Area Change

Infinity

…

Image source: wikipedia

The Derivative

1. Choose an interval 2. Find the raw change

3. Find the rate of change4. Make your model perfect

1 + 3 + 5 + 7 + 9 5x5

=

5 + 7 + 9 5x5

=

2x2-

George

Frank

g

f

f × dg

g × df

df × dgdg

df

Slicing A Cake Among Friends

Cake

A

A B C

New person?Cut a slice from everyone D

B A B

C

F’s changes

G’s changes+Simplifies to

Scenario With 2 Parts

F G+

F G*

F G^

F G/

…

Two Perspectives To Consider

Scenario With 3 Parts

A B× C×A’s

changesB’s

changes+C’s

changes+

3 Perspectives To Consider

x2x2

x2

Interaction Overall Change

Addition

Multiplication

Powers

Inverse

Division

Strategy Visualization Step-by-Step Layout Single Step Zoom

Ring-by-ringTimelapse

r dr

2πr

Symbolic Description Solution Notes

Work backwards to the integral.

that meansIf

Strategy Visualization Height of Plate Single Step Zoom

Plate-by-plateTimelapse

dx

π y2

x

yr

= + = 2 ×

Symbolic Solution Notes

Write height in terms of x

Work backwards to find integrals

Find volume at full radius (x=r)

Strategy Visualization Shell Analysis

Shell-by-shellX-Ray

Strategy Visualization Shell Analysis

Shell-by-shellX-Ray

dr

dV