Calculus Lesson 7 System Derivative ABC ABC ABC
![Calculus Lesson 7 Calculus Lesson 7](https://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-1.jpg)
Calculus Lesson 7
![System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] A^B / C []+[]+[] Three inputs, System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] A^B / C []+[]+[] Three inputs,](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-2.jpg)
System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] A^B / C []+[]+[] Three inputs, 3 changing perspectives to include
![George Frank George Frank](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-3.jpg)
George Frank
![dg f • dg df • dg g g • df f df dg f • dg df • dg g g • df f df](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-4.jpg)
dg f • dg df • dg g g • df f df
![Slicing A Cake Among Friends Cake A B C New person? Cut a slice Slicing A Cake Among Friends Cake A B C New person? Cut a slice](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-5.jpg)
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 Slicing A Cake Among Friends C Cake A A B New person? Cut a](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-6.jpg)
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 * Scenario With 3 Parts Change Simplifies To A + B + C A *](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-7.jpg)
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 ^ Scenario With 2 Parts F + G F * G Simplifies to F ^](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-8.jpg)
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 Scenario With 3 Parts A * B * C A’s changes + B’s changes](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-9.jpg)
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 Scenario With 2 Parts F + G F * G F’s changes Simplifies to](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-10.jpg)
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 F’s changes + Convert dg to dx Convert df to dx X’s changes G’s](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-11.jpg)
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 Hours Seconds/Hour Seconds G’s changes Convert dg to dx X’s changes + If you](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-12.jpg)
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 derivative of (x + 3) * (2 x + 7) derivative of = f](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-13.jpg)
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 derivative of (x 2 + 3 x + 1)6 derivative of = a 6](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-14.jpg)
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 Paint $ F’s changes df/dx x’s changes df/dx dx’s changes dg ----dx dx’s changes](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-15.jpg)
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 Wood $ + Convert Paint $ to ¥ Convert Wood $ to ¥ Wood](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-16.jpg)
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 F’s Changes + Convert dg to dx Convert df to dx X’s changes G’s](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-17.jpg)
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 System Derivative A+B+C []+[]+[] A*B*C []+[]+[] A^(B^C) []+[]+[] Three inputs, 3 changing perspectives to](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-18.jpg)
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 Scenario With 2 Parts System Fuzzy Viewpoint Derivative A + B A’s changes A*B*C](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-19.jpg)
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](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-20.jpg)
x 2 x 2
![x 2 x 2 x 2 x 2](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-21.jpg)
x 2 x 2
![](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-22.jpg)
![dg f * dg g g * df f df dg f * dg g g * df f df](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-23.jpg)
dg f * dg g g * df f df
![Calculus Week 8 Calculus Week 8](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-24.jpg)
Calculus Week 8
![Interaction Addition Multiplication Powers Inverse Division Overall Change Interaction Addition Multiplication Powers Inverse Division Overall Change](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-25.jpg)
Interaction Addition Multiplication Powers Inverse Division Overall Change
![Interaction Overall Change Analogy Addition Track changes from each part Multiplication Grow a rectangle Interaction Overall Change Analogy Addition Track changes from each part Multiplication Grow a rectangle](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-26.jpg)
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 X-Ray Strategy Visualization Step-by-Step Layout Step Zoom In Ring-by-ring 2 * pi * r](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-27.jpg)
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 Strategy Visualization Step-by-Step Layout Single Step Zoom Ring-by-ring Timelapse 2πr dr r Symbolic Description](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-28.jpg)
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 Strategy Plate-by-plate Timelapse Visualization Height of Plate r y Single Step Zoom π y](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-29.jpg)
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 Strategy Plate-by-plate Timelapse Visualization Height of Plate r y Single Step Zoom π y](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-30.jpg)
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 Symbolic Solution Notes Write height in terms of x Work backwards to find integrals](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-31.jpg)
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 - &= 2 int_0^r pi y^2 dx \ &= 2 int_0^r pi (sqrt{r^2 -](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-32.jpg)
&= 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 Strategy Visualization Shell Analysis Shell-by-shell X-Ray Strategy Shell-by-shell X-Ray Visualization Shell Analysis d. V](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-33.jpg)
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 Strategy Visualization Shell-by-shell X-Ray volume change / thickness change](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-34.jpg)
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 Symbolic Solution Notes Express height (y) in terms of x Work backwards to the](http://slidetodoc.com/presentation_image/835e27c65ff238ea04e542644f00a1f4/image-35.jpg)
Symbolic Solution Notes Express height (y) in terms of x Work backwards to the integral Get volume for full radius (x=r)
- Slides: 35