Derivative A B C A B C ABC AB C Three inputs 3 changing perspectives to include George Frank g f f ID: 791347
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Slide1
Calculus Lesson 7
Slide2System
Derivative
A + B + C
[ ] + [ ] + [ ]A * B * C[ ] + [ ] + [ ]A^(B^C)[ ] + [ ] + [ ]A^B / C[ ] + [ ] + [ ]
Three inputs, 3 changing perspectives to include
Slide3George
Frank
Slide4g
f
f
•
dg
g
•
df
df
•
dg
dg
df
Slide5Slicing A Cake Among Friends
A
B
C
A
B
C
New person?
Cut a slice from everyone
Cake
D
Slide6Slicing A Cake Among Friends
Cake
A
A
B
C
New person?
Cut a slice from everyone
D
B
A
B
C
Slide7A’s changes
B’s changes
C’s changes
+
+
Scenario With 3 Parts
Change Simplifies To
A
B
C
+
+
A
B
C
*
*
A
B
C
^
^
A
B
C
*
/
…
Slide8F’s changes
G’s changes
+
Simplifies to
Scenario With 2 Parts
F
G
+
F
G
*
F
G
^
F
G
/
…
Slide9Scenario With 3 Parts
A
B
*
C
*
A’s changes
B’s changes
+
C’s changes
+
Slide10Simplifies to
Scenario With 2 Parts
F
G
+
F
G
*
F
G
^
F
G
/
…
F’s changes
G’s changes
+
Convert df to dx
X’s changes
X’s changes
+
Convert dg to dx
Slide11F’s changes
G’s changes
+
X’s changesX’s changes
+
Convert dg to dx
Convert df to dx
Slide12Hours
G’s changes
X’s changes
X’s changes+
Convert dg to dx
Seconds
Seconds/Hour
[ f(g(x)) ]’ = f’(g(x)) * g’(x)
[ f(A) ]’ = f’(A) * A’
If you stop analyzing at A… then A’ = 1
dA
/
dA
= 1
df/dx = df/dg * dg/dx
dollars/yen = dollars/euro * euro/yen
Slide13f
g
*
derivative of derivative of
=
+
f
dg
*
df
g
*
=
dg/dx = 2
df/dx = 1
(x + 3)
(2x + 7)
2dx
1dx
(x + 3)
(2x + 7)
*
Slide14a
6
derivative of
derivative of =
a
=
da/dx = 2x + 3
(x
2
+ 3x + 1)
2dx
(x
2
+ 3x + 1)
6
5
6
Slide15x’s changes
F’s changes
df/dx
df’s changes
df/dx
dx’s
changes
dg’s changes
dg
-----
dx
dx’s
changes
Paint $
Slide16Wood $
Paint $
+
Wood ¥
Paint
¥
+
Convert Wood $ to
¥
Convert Paint $ to
¥
Slide17F’s Changes
G’s Changes
+
X’s changesX’s changes
+
Convert df to dx
Convert dg to dx
Slide18System
Derivative
A + B + C
[ ] + [ ] + [ ]A * B * C[ ] + [ ] + [ ]A^(B^C)[ ] + [ ] + [ ][ ] + [ ] + [ ]
Three inputs, 3 changing perspectives to include
Slide19System
Derivative
Fuzzy Derivative
A * B * C[ ] + [ ] + [ ]A^(B^C)[ ] + [ ] + [ ][ ] + [ ] + [ ]
Scenario With 2 Parts
Fuzzy Viewpoint
A
B
+
…
A’s changes
B’s changes
+
Slide20x
2
x
2
x
2
Slide21x
2
x
2x2
Slide22Slide23g
f
f
* dg
g * df
dg
df
Slide24Calculus Week 8
Slide25Interaction
Overall
Change
AdditionMultiplicationPowersInverseDivision
Slide26Interaction
Overall
Change
AnalogyAdditionTrack changes from each partMultiplicationGrow a rectanglePowers
N viewpoints of
“my change times the others”
Inverse
Sharing
cake, new guy walks in
Division
Imagine
f * (1/g)
Slide27X-Ray
Strategy
Visualization
Step-by-Step LayoutStep Zoom InRing-by-ring
r
dr
Symbolic
Solution
Step Zoom
In
r
dr
(from 0 to r)
2 * pi * r
Slide28Strategy
Visualization
Step-by-Step Layout
Single Step ZoomRing-by-ringTimelapse
r
dr
2
π
r
Symbolic
Description
Solution
Notes
Work backwards to the integral.
that means
If
Slide29Strategy
Visualization
Height of Plate
Single Step ZoomPlate-by-plateTimelapse
dx
π
y
2
x
y
r
Slide30Strategy
Visualization
Height of Plate
Single Step ZoomPlate-by-plateTimelapse
x
dx
π
y
2
x
y
r
Slide31Symbolic
Solution
Notes
Write height in terms of x
Work backwards to find integrals
Find volume at full radius (x=r)
Slide32&= 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
Slide33Strategy
Visualization
Shell Analysis
Shell-by-shellX-Ray
Strategy
Visualization
Shell Analysis
Shell-by-shell
X-Ray
dr
dV
Slide34Strategy
Visualization
Shell-by-shell
X-Ray
volume change
/
thickness change
Slide35Symbolic
Solution
Notes
Express height (y) in terms of x
Work backwards to the integral
Get volume for full radius (x=r)