A Guide To Calculus Baseline Rendering Baseline vs
A Guide To Calculus
Baseline Rendering
Baseline vs. Progressive Rendering 25% 50% 100%
Circle and Sphere Fun Facts
Dissecting a Circle ~ Disc Rings
Unroll the Rings Area = ½ base x height
X-Ray Original Rings Slices Boards
Ring By Ring 25% 50% 75% 100%
Slice By Slice
Board By Board
Strategy Visualization Step-by-Step Layout Single Step Zoom Ring-by-ring r dr Slice-by-slice p dp Board-by-board x dx
Strategy Ring-by-ring Slice-by-slice Board-by-board Visualization Step-by-Step Layout Calculus Lingo
Circle Rings Slices Boards Sphere Shells Wedges Plates
Circumference Ring-by-ring Timelapse Area Volume Plate-by-plate Timelapse Surface area Shell-by-Shell X-Ray
Derivative Symbol Mindset Think about the resulting step-by -step pattern Think about the actual ratio of changes
1 d Perimeter Change 2 d Area Change x 1 x
Infinity … Image source: wikipedia
The Derivative 1. Choose an interval 4. Make your model perfect 2. Find the raw change 3. Find the rate of change
= 1 +3 +5 +7 +9 5 x 5
= 5 +7 +9 5 x 5 - 2 x 2
George Frank
dg f × dg df × dg g g × df f df
Slicing A Cake Among Friends C Cake A A B New person? Cut a slice from everyone D A B C B
Scenario With 2 Parts F + G F * G Two Perspectives To Consider Simplifies to F ^ G F / … G F’s changes + G’s changes
Scenario With 3 Parts A × B × C 3 Perspectives To Consider A’s changes + B’s changes + C’s changes
x 2 x 2
Interaction Addition Multiplication Powers Inverse Division Overall Change
Strategy Visualization Step-by-Step Layout Single Step Zoom Ring-by-ring Timelapse 2πr dr r Symbolic Description Solution Notes Work backwards to the integral. If that means
Strategy Plate-by-plate Timelapse Visualization Height of Plate r y Single Step Zoom π y 2 x dx
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-shell X-Ray Strategy Shell-by-shell X-Ray Visualization Shell Analysis d. V dr
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