daviddoji/CalculusWithJuliaNotes.jl
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use quarto, not Pluto to render pages

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jverzani
2022-07-24 16:38:24 -04:00
parent 93c993206a
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CwJ/derivatives/lhospitals_rule.jmd
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@@ -15,7 +15,7 @@ using SymPy
using CalculusWithJulia.WeaveSupport
using Roots
fig_size=(600, 400)
fig_size=(800, 600)
const frontmatter = (
title = "L'Hospital's Rule",
description = "Calculus with Julia: L'Hospital's Rule",
@@ -79,10 +79,10 @@ likely due to one of the Bernoulli brothers.
> ``\lim_{x \rightarrow c+}f(x)/g(x) = L``.
That is *if* the right limit of ``f(x)/g(x)`` is indeterminate of the form ``0/0``,
but the right limit of ``f'(x)/g'(x)`` is known, possibly by simple
continuity, then the right limit of ``f(x)/g(x)`` exists and is equal to that
of ``f'(x)/g'(x)``.
That is *if* the right limit of ``f(x)/g(x)`` is indeterminate of the
form ``0/0``, but the right limit of ``f'(x)/g'(x)`` is known,
possibly by simple continuity, then the right limit of ``f(x)/g(x)``
exists and is equal to that of ``f'(x)/g'(x)``.
The rule equally applies to *left limits* and *limits* at ``c``. Later it will see there are other generalizations.
@@ -102,12 +102,8 @@ this answer is as it is, but we don't need to think in terms of
\approx x``, as ``\cos(0)`` appears as the coefficient.
```julia; echo=false
note("""
In [Gruntz](http://www.cybertester.com/data/gruntz.pdf), in a reference attributed to Speiss, we learn that L'Hospital was a French Marquis who was taught in ``1692`` the calculus of Leibniz by Johann Bernoulli. They made a contract obliging Bernoulli to leave his mathematical inventions to L'Hospital in exchange for a regular compensation. This result was discovered in ``1694`` and appeared in L'Hospital's book of ``1696``.
"""; title="Bernoulli-de l'Hospital")
```
!!! note
In [Gruntz](http://www.cybertester.com/data/gruntz.pdf), in a reference attributed to Speiss, we learn that L'Hospital was a French Marquis who was taught in ``1692`` the calculus of Leibniz by Johann Bernoulli. They made a contract obliging Bernoulli to leave his mathematical inventions to L'Hospital in exchange for a regular compensation. This result was discovered in ``1694`` and appeared in L'Hospital's book of ``1696``.
##### Examples
@@ -121,10 +117,8 @@ In [Gruntz](http://www.cybertester.com/data/gruntz.pdf), in a reference attribut
= \lim_{x \rightarrow 0}\frac{a^x - 1}{x}.
```
```julia; echo=false
note("""Why rewrite in the "opposite" direction? Because the theorem's result -- ``L`` is the limit -- is only true if the related limit involving the derivative exists. We don't do this in the following, but did so here to emphasize the need for the limit of the ratio of the derivatives to exist.
""")
```
!!! note
Why rewrite in the "opposite" direction? Because the theorem's result -- ``L`` is the limit -- is only true if the related limit involving the derivative exists. We don't do this in the following, but did so here to emphasize the need for the limit of the ratio of the derivatives to exist.
- Consider this limit:
@@ -270,7 +264,8 @@ known.
----
```julia; hold=true; echo=false; cache=true
```julia; echo=false; cache=true
let
## {{{lhopitals_picture}}}
function lhopitals_picture_graph(n)
@@ -286,8 +281,8 @@ function lhopitals_picture_graph(n)
## get bounds
tl = (x) -> g(0) + m * (x - f(0))
lx = max(fzero(x -> tl(x) - (-0.05),-1000, 1000), -0.6)
rx = min(fzero(x -> tl(x) - (0.25),-1000, 1000), 0.2)
lx = max(find_zero(x -> tl(x) - (-0.05), (-1000, 1000)), -0.6)
rx = min(find_zero(x -> tl(x) - (0.25), (-1000, 1000)), 0.2)
xs = [lx, rx]
ys = map(tl, xs)
@@ -319,7 +314,8 @@ gif(anim, imgfile, fps = 1)
plotly()
ImageFile(imgfile, caption)
ImageFile(imgfile, caption)
end
```
## Generalizations
@@ -556,8 +552,8 @@ nothing
```
```julia; hold=true; echo=false
ans = 1
radioq(lh_choices, ans, keep_order=true)
answ = 1
radioq(lh_choices, answ, keep_order=true)
```
###### Question
@@ -565,8 +561,8 @@ radioq(lh_choices, ans, keep_order=true)
This function ``f(x) = \sin(x)^{\sin(x)}`` is *indeterminate* at ``x=0``. What type?
```julia; hold=true; echo=false
ans =3
radioq(lh_choices, ans, keep_order=true)
answ =3
radioq(lh_choices, answ, keep_order=true)
```
###### Question
@@ -574,8 +570,8 @@ radioq(lh_choices, ans, keep_order=true)
This function ``f(x) = (x-2)/(x^2 - 4)`` is *indeterminate* at ``x=2``. What type?
```julia; hold=true; echo=false
ans = 1
radioq(lh_choices, ans, keep_order=true)
answ = 1
radioq(lh_choices, answ, keep_order=true)
```
###### Question
@@ -583,8 +579,8 @@ radioq(lh_choices, ans, keep_order=true)
This function ``f(x) = (g(x+h) - g(x-h)) / (2h)`` (``g`` is continuous) is *indeterminate* at ``h=0``. What type?
```julia; hold=true; echo=false
ans = 1
radioq(lh_choices, ans, keep_order=true)
answ = 1
radioq(lh_choices, answ, keep_order=true)
```
###### Question
@@ -592,8 +588,8 @@ radioq(lh_choices, ans, keep_order=true)
This function ``f(x) = x \log(x)`` is *indeterminate* at ``x=0``. What type?
```julia; hold=true; echo=false
ans = 5
radioq(lh_choices, ans, keep_order=true)
answ = 5
radioq(lh_choices, answ, keep_order=true)
```
@@ -610,8 +606,8 @@ choices = [
"Yes. It is of the form ``0/0``",
"No. It is not indeterminate"
]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
###### Question
@@ -769,6 +765,6 @@ choices = [
"``0``",
"It does not exist"
]
ans =1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```