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jverzani
2022-07-25 20:51:08 -04:00
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28
quarto/derivatives/curve_sketching.qmd
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@@ -1,33 +1,7 @@
# Curve Sketching
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses the following add-on packages:

28
quarto/derivatives/derivatives.qmd
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@@ -1,33 +1,7 @@
# Derivatives
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

28
quarto/derivatives/first_second_derivatives.qmd
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@@ -1,33 +1,7 @@
# The first and second derivatives
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

28
quarto/derivatives/implicit_differentiation.qmd
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@@ -1,33 +1,7 @@
# Implicit Differentiation
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

28
quarto/derivatives/lhospitals_rule.qmd
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@@ -1,33 +1,7 @@
# L'Hospital's Rule
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

28
quarto/derivatives/linearization.qmd
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@@ -1,33 +1,7 @@
# Linearization
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

62
quarto/derivatives/mean_value_theorem.qmd
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@@ -1,33 +1,7 @@
# The mean value theorem for differentiable functions.
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:
@@ -390,7 +364,39 @@ That is the function $f(x)$, minus the secant line between $(a,f(a))$ and $(b, f
nothing
```
An interactive example can be found at [jsxgraph](http://jsxgraph.uni-bayreuth.de/wiki/index.php?title=Mean_Value_Theorem).
```{=html}
<div id="jsxgraph" style="width: 500px; height: 500px;"></div>
```
```{ojs}
//| echo: false
//| output: false
JXG = require("jsxgraph");
board = JXG.JSXGraph.initBoard('jsxgraph', {boundingbox: [-5, 10, 7, -6], axis:true});
p = [
board.create('point', [-1,-2], {size:2}),
board.create('point', [6,5], {size:2}),
board.create('point', [-0.5,1], {size:2}),
board.create('point', [3,3], {size:2})
];
f = JXG.Math.Numerics.lagrangePolynomial(p);
graph = board.create('functiongraph', [f,-10, 10]);
g = function(x) {
return JXG.Math.Numerics.D(f)(x)-(p[1].Y()-p[0].Y())/(p[1].X()-p[0].X());
};
r = board.create('glider', [
function() { return JXG.Math.Numerics.root(g,(p[0].X()+p[1].X())*0.5); },
function() { return f(JXG.Math.Numerics.root(g,(p[0].X()+p[1].X())*0.5)); },
graph], {name:' ',size:4,fixed:true});
board.create('tangent', [r], {strokeColor:'#ff0000'});
line = board.create('line',[p[0],p[1]],{strokeColor:'#ff0000',dash:1});
```
This interactive example can also be found at [jsxgraph](http://jsxgraph.uni-bayreuth.de/wiki/index.php?title=Mean_Value_Theorem). It shows a cubic polynomial fit to the $4$ adjustable points labeled A through D. The secant line is drawn between points A and B with a dashed line. A tangent line – with the same slope as the secant line – is identified at a point $(\alpha, f(\alpha))$ where $\alpha$ is between the points A and B. That this can always be done is a conseuqence of the mean value theorem.
##### Example

28
quarto/derivatives/more_zeros.qmd
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@@ -1,33 +1,7 @@
# Derivative-free alternatives to Newton's method
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

119
quarto/derivatives/newtons_method.qmd
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@@ -1,33 +1,7 @@
# Newton's method
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:
@@ -311,6 +285,97 @@ gif(anim, imgfile, fps = 1)
ImageFile(imgfile, caption)
```
---
This interactive graphic (built using [JSXGraph](https://jsxgraph.uni-bayreuth.de/wp/index.html)) allows the adjustment of the point `x0`, initially at $0.85$. Five iterations of Newton's method are illustrated. Different positions of `x0` clearly converge, others will not.
```{=html}
<div id="jsxgraph" style="width: 500px; height: 500px;"></div>
```
```{ojs}
//| echo: false
//| output: false
JXG = require("jsxgraph");
// newton's method
b = JXG.JSXGraph.initBoard('jsxgraph', {
boundingbox: [-3,5,3,-5], axis:true
});
f = function(x) {return x*x*x*x*x - x - 1};
fp = function(x) { return 4*x*x*x*x - 1};
x0 = 0.85;
nm = function(x) { return x - f(x)/fp(x);};
l = b.create('point', [-1.5,0], {name:'', size:0});
r = b.create('point', [1.5,0], {name:'', size:0});
xaxis = b.create('line', [l,r])
P0 = b.create('glider', [x0,0,xaxis], {name:'x0'});
P0a = b.create('point', [function() {return P0.X();},
function() {return f(P0.X());}], {name:''});
P1 = b.create('point', [function() {return nm(P0.X());},
0], {name:''});
P1a = b.create('point', [function() {return P1.X();},
function() {return f(P1.X());}], {name:''});
P2 = b.create('point', [function() {return nm(P1.X());},
0], {name:''});
P2a = b.create('point', [function() {return P2.X();},
function() {return f(P2.X());}], {name:''});
P3 = b.create('point', [function() {return nm(P2.X());},
0], {name:''});
P3a = b.create('point', [function() {return P3.X();},
function() {return f(P3.X());}], {name:''});
P4 = b.create('point', [function() {return nm(P3.X());},
0], {name:''});
P4a = b.create('point', [function() {return P4.X();},
function() {return f(P4.X());}], {name:''});
P5 = b.create('point', [function() {return nm(P4.X());},
0], {name:'x5', strokeColor:'black'});
P0a.setAttribute({fixed:true});
P1.setAttribute({fixed:true});
P1a.setAttribute({fixed:true});
P2.setAttribute({fixed:true});
P2a.setAttribute({fixed:true});
P3.setAttribute({fixed:true});
P3a.setAttribute({fixed:true});
P4.setAttribute({fixed:true});
P4a.setAttribute({fixed:true});
P5.setAttribute({fixed:true});
sc = '#000000';
b.create('segment', [P0,P0a], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P0a, P1], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P1,P1a], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P1a, P2], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P2,P2a], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P2a, P3], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P3,P3a], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P3a, P4], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P4,P4a], {strokeColor:sc, strokeWidth:1});
b.create('segment', [P4a, P5], {strokeColor:sc, strokeWidth:1});
b.create('functiongraph', [f, -1.5, 1.5])
```
##### Example: numeric not algebraic

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quarto/derivatives/numeric_derivatives.qmd
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@@ -1,33 +1,7 @@
# Numeric derivatives
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

28
quarto/derivatives/optimization.qmd
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@@ -1,33 +1,7 @@
# Optimization
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:

44
quarto/derivatives/process.jl
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@@ -1,44 +0,0 @@
using WeavePynb
using CwJWeaveTpl
fnames = [
"derivatives", ## more questions
"numeric_derivatives",
"mean_value_theorem",
"optimization",
"curve_sketching",
"linearization",
"newtons_method",
"lhopitals_rule", ## Okay - -but could beef up questions..
"implicit_differentiation", ## add more questions?
"related_rates",
"taylor_series_polynomials"
]
process_file(nm; cache=:off) = CwJWeaveTpl.mmd(nm * ".jmd", cache=cache)
function process_files(;cache=:user)
for f in fnames
@show f
process_file(f, cache=cache)
end
end
"""
## TODO derivatives
tangent lines intersect at avearge for a parabola
Should we have derivative results: inverse functions, logarithmic differentiation...
"""

28
quarto/derivatives/related_rates.qmd
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@@ -1,33 +1,7 @@
# Related rates
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packaages:

28
quarto/derivatives/symbolic_derivatives.qmd
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@@ -1,33 +1,7 @@
# Symbolic derivatives
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses this add-on package:

28
quarto/derivatives/taylor_series_polynomials.qmd
View File

@@ -1,33 +1,7 @@
# Taylor Polynomials and other Approximating Polynomials
```{julia}
#| echo: false
import Logging
Logging.disable_logging(Logging.Info) # or e.g. Logging.Info
Logging.disable_logging(Logging.Warn)
import SymPy
function Base.show(io::IO, ::MIME"text/html", x::T) where {T <: SymPy.SymbolicObject}
println(io, "<span class=\"math-left-align\" style=\"padding-left: 4px; width:0; float:left;\"> ")
println(io, "\\[")
println(io, sympy.latex(x))
println(io, "\\]")
println(io, "</span>")
end
# hack to work around issue
import Markdown
import CalculusWithJulia
function CalculusWithJulia.WeaveSupport.ImageFile(d::Symbol, f::AbstractString, caption; kwargs...)
nm = joinpath("..", string(d), f)
u = "![$caption]($nm)"
Markdown.parse(u)
end
nothing
```
{{< include ../_common_code.qmd >}}
This section uses these add-on packages: