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Copy pathpluto_differentiability backup 1.jl
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pluto_differentiability backup 1.jl
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### A Pluto.jl notebook ###
# v0.19.42
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 5ffc5ec0-f91f-11ec-0552-37ef5f25102d
begin
using Symbolics
using ForwardDiff
using Plots
using PlutoUI
using LaTeXStrings
end
# ╔═╡ 2dc43448-3cf4-475f-8b61-d16c3594d9e3
md"""
## Differentiability
Let $f(x)$ be a function defined near $x=a$ in $\mathbb{R}$. We say that $f(x)$ is differentiable at $x=a$ if there exists $\alpha\in\mathbb{R}$ such that
$\frac{f(x)-f(a)}{x-a} \rightarrow \alpha \quad (x \rightarrow a).$
In this case the real number $\alpha$ is said to be the derivative of $f(x)$ at $x=a$ and denoted by
$f^\prime(a), \quad \frac{df}{dx}(a).$
Set
$R(x):=f(x)-\{f(a)+\alpha(x-a)\},$
that is,
$f(x)=f(a)+\alpha(x-a)+R(x), \quad \frac{R(x)}{x-a}=\frac{f(x)-f(a)}{x-a} - \alpha.$
Then $f(x)$ is differentiable at $x=a$ if and only if there exist $\alpha\in\mathbb{R}$ and a function $R(x)$ such that
$f(x)={\color{blue}{f(a)+\alpha(x-a)}}+R(x), \quad {\color{red}{\frac{R(x)}{x-a} \rightarrow 0}} \quad (x \rightarrow a).$
Let us consider the meaning of the differentiability below.
"""
# ╔═╡ 06191b8b-ddcc-4a2b-8228-0680aa8ad7c5
md"""
## About $\quad {\color{red}{R(x)/(x-a) \rightarrow 0}} \quad (x \rightarrow a)$
The condition ${\color{red}{\dfrac{R(x)}{x-a} \rightarrow 0}} \quad (x \rightarrow a)$ means that $R(x)$ goes to $0$ as $x \rightarrow a$ faster than the linear funtion of $x-a$.
"""
# ╔═╡ 4c26a3e2-7ad7-4ca5-a5bb-1cf747f1a11a
md"""
p = $(@bind r Slider(0.1:0.1:10, show_value=true, default=2))
"""
# ╔═╡ 8d9148e7-4ce1-4da7-9eb3-43df098d5ad5
begin
p0=zeros(2)
q0=zeros(2)
p0[1]=1
p0[2]=101
q0[1]=0
q0[2]=1
t = range(0, 1.2, length = 121)
supersub=zeros(121)
linear=zeros(121)
for k=1:121
supersub[k]=t[k]^r
linear[k]=t[k]
end
end
# ╔═╡ 769adba4-34e2-4e63-9099-2df3f6dbb529
begin
plot([supersub,linear],grid=false,
linewidth=2,ylim=(-0.1,1.5),
title="s=t^p and s=t",
xticks = ([0 101;], [0 1]),
xlabel="t",
yticks = ([0 1;], [0 1]),
ylabel="s",
label = ["s=t^p" "s=t"],
legend = :topleft)
scatter!(p0,q0,markersize = 5,label = false)
end
# ╔═╡ a488a14c-68b0-45d7-9b31-2aef8f394471
md"""
## All the lines passing through $\bigl(a,f(a)\bigr)$
Consider lines given by the graph of functions
$g(x)={\color{darkorange}{f(a)+\beta(x-a)}}, \quad \beta\in\mathbb{R}, \qquad h(x)={\color{green}{f(a)+f^\prime(a)(x-a)}}.$
$h(x)$ is the special case of $g(x)$ with $\beta=f^\prime(a)$. Consider how fast the differences $f(x)-g(x)$ and $f(x)-h(x)$ go to $0$ as $x \rightarrow a$.
- If $\beta \ne f^\prime(a)$ then as fast as $x-a$.
$f(x)-g(x)={\color{magenta}{\bigl(f^\prime(a)-\beta\bigr)(x-a)}} + {\color{red}{R(x)}}.$
- If $\beta = f^\prime(a)$ then faster than $x-a$.
$f(x)-h(x)={\color{red}{R(x)}}.$
So the line given by $y=h(x)$ is a special line for $y=f(x)$ at $\bigl(a,f(a)\bigr)$. The line given by $y=h(x)$ is said to be the tangential line of $y=f(x)$ at $\bigl(a,f(a)\bigr)$. Roughly spaeking, we say that $f(x)$ is differentiable at $x=a$ if there exists a unique tangential line of $y=f(x)$ at $\bigl(a,f(a)\bigr)$.
"""
# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
#### Example $\quad f(x)=x^2$
$g(x)={\color{darkorange}{a^2+\beta(x-a)}},
\quad
h(x)={\color{green}{a^2+2a(x-a)}}.$
$f(x)-g(x)={\color{magenta}{(2a-\beta)(x-a)}}+{\color{red}{(x-a)^2}}, \quad f(x)-h(x)={\color{red}{(x-a)^2}}.$
"""
# ╔═╡ 092ba07e-4866-40c6-ade8-25d9c2218011
md"""
a = $(@bind a1 Slider(-4:0.1:4, show_value=true, default=1))
β = $(@bind b1 Slider(-40:0.1:40, show_value=true, default=1))
"""
# ╔═╡ ab1c3d69-0499-4fad-b2a8-5360ed7339e1
begin
p1=zeros(1)
q1=zeros(1)
p1[1]=501+100*a1
q1[1]=a1^2
x1 = range(-5, 5, length = 1001)
f1=zeros(1001)
g1=zeros(1001)
h1=zeros(1001)
for k=1:1001
f1[k]=x1[k]^2
g1[k]=b1*(x1[k]-a1)+a1^2
h1[k]=2*a1*(x1[k]-a1)+a1^2
end
end
# ╔═╡ 8ed1312b-f5a3-4220-a865-9a12ed09a691
begin
plot([f1,g1,h1],grid=false,
linewidth=2,ylim=(-2,22),
title="y=f(x) and all the lines passing through (a,f(a))",
xticks = ([0 501 1001;], [-5 0 5]),
xlabel="x",
label = ["y=f(x)" "y=g(x)" "y=h(x)"],
ylabel="y",
legend = :top)
scatter!(p1,q1,markersize = 5,label = false)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
[compat]
ForwardDiff = "~0.10.30"
LaTeXStrings = "~1.3.0"
Plots = "~1.31.1"
PlutoUI = "~0.7.39"
Symbolics = "~4.8.3"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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