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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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-
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
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
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