Calculus Lesson 7 System Derivative ABC ABC ABC

Calculus Lesson 7

System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] A^B / C []+[]+[] Three inputs, 3 changing perspectives to include

George Frank

dg f • dg df • dg g g • df f df

Slicing A Cake Among Friends Cake A B C New person? Cut a slice from everyone D A B C

Slicing A Cake Among Friends C Cake A A B New person? Cut a slice from everyone D A B C B

Scenario With 3 Parts Change Simplifies To A + B + C A * B * C A ^ B ^ C A * B / … C A’s changes + B’s changes + C’s changes

Scenario With 2 Parts F + G F * G Simplifies to F ^ G F / … G F’s changes + G’s changes

Scenario With 3 Parts A * B * C A’s changes + B’s changes + C’s changes

Scenario With 2 Parts F + G F * G F’s changes Simplifies to F ^ G F / … G + Convert df to dx X’s changes G’s changes Convert dg to dx + X’s changes

F’s changes + Convert dg to dx Convert df to dx X’s changes G’s changes + X’s changes

Hours Seconds/Hour Seconds G’s changes Convert dg to dx X’s changes + If you stop analyzing a [ f(g(x)) ]’ = f’(g(x)) * g’(x) [ f(A) ]’ = f’(A) * A’ df/dx = df/dg * dg/dx dollars/yen = dollars/euro * euro/yen d. A/d. A = 1

derivative of (x + 3) * (2 x + 7) derivative of = f * g = f * dg dg/dx = 2 (x + 3) 2 dx + g * df df/dx = 1 (2 x + 7) 1 dx

derivative of (x 2 + 3 x + 1)6 derivative of = a 6 = 6 a 5 da/dx = 2 x + 3 (x 2 + 3 x + 1) 2 dx

Paint $ F’s changes df/dx x’s changes df/dx dx’s changes dg ----dx dx’s changes

Wood $ + Convert Paint $ to ¥ Convert Wood $ to ¥ Wood ¥ Paint $ + Paint ¥

F’s Changes + Convert dg to dx Convert df to dx X’s changes G’s Changes + X’s changes

System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] Three inputs, 3 changing perspectives to include

Scenario With 2 Parts System Fuzzy Viewpoint Derivative A + B A’s changes A*B*C []+[]+[] A^(B^C) []+[]+[] … Fuzzy Derivative + B’s changes

x 2 x 2

x 2 x 2

dg f * dg g g * df f df

Calculus Week 8

Interaction Addition Multiplication Powers Inverse Division Overall Change

Interaction Overall Change Analogy Addition Track changes from each part Multiplication Grow a rectangle Powers N viewpoints of “my change times the others” Inverse Sharing cake, new guy walks in Division Imagine f * (1/g)

X-Ray Strategy Visualization Step-by-Step Layout Step Zoom In Ring-by-ring 2 * pi * r r Symbolic Solution dr (from 0 to r) Step Zoom In r dr

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

Strategy Plate-by-plate Timelapse Visualization Height of Plate r y Single Step Zoom π y 2 x dx x

Symbolic Solution Notes Write height in terms of x Work backwards to find integrals Find volume at full radius (x=r)

&= 2 int_0^r pi y^2 dx \ &= 2 int_0^r pi (sqrt{r^2 - x^2})^2 dx \ &= 2 pi int_0^r r^2 - x^2 dx \ &= 2 pi left( (r^2)x - frac{1}{3}x^3 right) \ &= 2 pi left( (r^2)r - frac{1}{3}r^3 right) \ &= 2 pi left( frac{2}{3}r^3 right) \ &= frac{4}{3} pi r^3

Strategy Visualization Shell Analysis Shell-by-shell X-Ray Strategy Shell-by-shell X-Ray Visualization Shell Analysis d. V dr

Strategy Visualization Shell-by-shell X-Ray volume change / thickness change

Symbolic Solution Notes Express height (y) in terms of x Work backwards to the integral Get volume for full radius (x=r)