calculus lesson 7
DESCRIPTION
Calculus Lesson 7. Three inputs, 3 changing perspectives to include. George. Frank. dg. f • dg. df • dg. g. g • df. df. f. Slicing A Cake Among Friends. A. B. C. Cake. New person? Cut a slice from everyone. D. A. B. C. Slicing A Cake Among Friends. A. A. B. Cake. B. - PowerPoint PPT PresentationTRANSCRIPT
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
g
f
f • dg
g • df
df • dgdg
df
Slicing A Cake Among Friends
A B C
A B C
New person?Cut a slice from everyone
Cake
D
Slicing A Cake Among Friends
Cake
A
A B C
New person?Cut a slice from everyone D
B A B
C
A’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* /
…
F’s changes
G’s changes+Simplifies to
Scenario With 2 Parts
F G+
F G*
F G^
F G/
…
Scenario With 3 Parts
A B* C*A’s
changesB’s
changes+C’s
changes+
Simplifies 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
F’s changes
G’s changes+
X’s changes
X’s changes+
Convert dg to dxConvert df to dx
Hours G’s changes
X’s changes
X’s changes+
Convert dg to dx
SecondsSeconds/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 = 1df/dx = df/dg * dg/dx
dollars/yen = dollars/euro * euro/yen
f g*
derivative of derivative of
= +f dg* dfg *=
dg/dx = 2
df/dx = 1
(x + 3) (2x + 7)2dx 1dx
(x + 3) (2x + 7)*
a 6
derivative of derivative of
= a=
da/dx = 2x + 3
(x2 + 3x + 1) 2dx
(x2 + 3x + 1)6 56
x’s changes
F’s changes df/dx
df’s changes df/dx dx’s
changes
dg’s changes
dg-----dx
dx’schanges
Paint $
Wood $ Paint $+
Wood ¥ Paint ¥+
Convert Wood $ to ¥ Convert Paint $ to ¥
F’s Changes
G’s Changes+
X’s changes
X’s changes+
Convert df to dx Convert dg to dx
System Derivative
A + B + C [ ] + [ ] + [ ]
A * B * C [ ] + [ ] + [ ]
A^(B^C) [ ] + [ ] + [ ]
[ ] + [ ] + [ ]
Three inputs, 3 changing perspectives to include
System Derivative Fuzzy Derivative
A * B * C [ ] + [ ] + [ ]
A^(B^C) [ ] + [ ] + [ ]
[ ] + [ ] + [ ]
Scenario With 2 Parts Fuzzy Viewpoint
A B+
…
A’s changes
B’s changes
+
x2 x2
x2
x2x2
x2
g
f
f * dg
g * df
dg
df
Calculus Week 8
Interaction Overall Change
Addition
Multiplication
Powers
Inverse
Division
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
rdr
Symbolic Solution Step Zoom In
r dr
(from 0 to r)
2 * pi * r
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
Strategy Visualization Height of Plate Single Step Zoom
Plate-by-plateTimelapse
x
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)
&= 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-shellX-Ray
Strategy Visualization Shell Analysis
Shell-by-shellX-Ray
dr
dV
Strategy Visualization
Shell-by-shellX-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)