work on limits section

quarto/_quarto.yml

@@ -1,4 +1,4 @@
-version: "0.20"
+version: "0.21"
 engine: julia
 
 project:
@@ -23,7 +23,7 @@ book:
     pinned: false
   sidebar:
     collapse-level: 1
-  page-footer: "Copyright 2022, John Verzani"
+  page-footer: "Copyright 2022-24, John Verzani"
   chapters:
     - index.qmd
     - part: precalc.qmd

quarto/index.qmd

@@ -68,11 +68,25 @@ and loads some useful packages that will be used repeatedly.
 These notes are presented as a Quarto book. To learn more about Quarto
 books visit <https://quarto.org/docs/books>.
 
+<!--
 These notes may be compiled into a `pdf` file through Quarto. As the result is rather large, we do not provide that file for download. For the interested reader, downloading the repository, instantiating the environment, and running `quarto` to render to `pdf` in the `quarto` subdirectory should produce that file (after some time).
+-->
 
 To *contribute* -- say by suggesting addition topics, correcting a
 mistake, or fixing a typo -- click the "Edit this page" link and join the list of [contributors](https://github.com/jverzani/CalculusWithJuliaNotes.jl/graphs/contributors). Thanks to all contributors and a *very* special thanks to `@fangliu-tju` for their careful and most-appreciated proofreading.
 
+## Running Julia
+
+`Julia` is installed quite easily with the `juliaup` utility. There are some brief installation notes in the overview of `Julia` commands. To run `Julia` through the web (though in a resource-constrained manner), these links resolve to `binder.org` instances:
+
+
+* [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/jverzani/CalculusWithJuliaBinder.jl/main?labpath=blank-notebook.ipynb) (Image without SymPy)
+
+
+* [![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/jverzani/CalculusWithJuliaBinder.jl/sympy?labpath=blank-notebook.ipynb) (Image with SymPy, longer to load)
+
+
+
 ----
 
 Calculus with Julia version {{< meta version >}}, produced on {{< meta date >}}.

quarto/integrals/area.qmd

@@ -974,7 +974,7 @@ let
 	F = FnWrapper(f)
 	ans,err = quadgk(F, a, b)
 	plot(f, a, b, legend=false, title="Error ≈ $(round(err,sigdigits=2))")
-			scatter!(F.xs, F.ys)
+	scatter!(F.xs, F.ys)
 end
 ```
 
@@ -1047,6 +1047,23 @@ solve.(Z, (1/4, 1/2, 3/4))
 
 The middle one is clearly $0$. This distribution is symmetric about $0$, so half the area is to the right of $0$ and half to the left, so clearly when $p=0.5$, $x$ is $0$. The other two show that the area to the left of $-0.809767$ is equal to the area to the right of $0.809767$ and equal to $0.25$.
 
+##### Example: Gauss nodes
+
+The `QuadGK.gauss(n)` function returns a pair of $n$ quadrature points and weights to integrate a function over the interval $(-1,1)$, with an option to use a different interval $(a,b)$. For a given $n$, these values exactly integrate any polynomial of degree $2n-1$ or less. The pattern to integrate below can be expressed in other ways, but this is intended to be direct:
+
+
+```{julia}
+xs, ws = QuadGK.gauss(5)
+```
+
+```{julia}
+f(x) = exp(cos(x))
+sum(w * f(x) for (x, w) in zip(xs, ws))
+```
+
+The `zip` function is used to iterate over the `xs` and `ws` as pairs of values.
+
+
 
 ## Questions
 

quarto/limits/continuity.qmd

@@ -76,6 +76,8 @@ This speaks to continuity at a point, we can extend this to continuity over an i
 
 Finally, as with limits, it can be convenient to speak of *right* continuity and *left* continuity at a point, where the limit in the definition is replaced by a right or left limit, as appropriate.
 
+In particular, a function is *continuous* over $[a,b]$ if it is continuous on $(a,b)$, left continuous at $b$ and right continuous at $a$.
+
 
 :::{.callout-warning}
 ## Warning
@@ -90,7 +92,7 @@ Most familiar functions are continuous everywhere.
 
 
   * For example, a monomial function $f(x) = ax^n$ for non-negative, integer $n$ will be continuous. This is because the limit exists everywhere, the domain of $f$ is all $x$ and there are no jumps.
-  * Similarly, the basic trigonometric functions $\sin(x)$, $\cos(x)$ are continuous everywhere.
+  * Similarly, the building-block trigonometric functions $\sin(x)$, $\cos(x)$ are continuous everywhere.
   * So are the exponential functions $f(x) = a^x, a > 0$.
   * The hyperbolic sine ($(e^x - e^{-x})/2$) and cosine ($(e^x + e^{-x})/2$) are, as $e^x$ is.
   * The hyperbolic tangent is, as $\cosh(x) > 0$ for all $x$.
@@ -103,7 +105,7 @@ Some familiar functions are *mostly* continuous but not everywhere.
   * Similarly, $f(x) = \log(x)$ is continuous on $(0,\infty)$, but it is not defined at $x=0$, so is not right continuous at $0$.
   * The tangent function $\tan(x) = \sin(x)/\cos(x)$ is continuous everywhere *except* the points $x$ with $\cos(x) = 0$ ($\pi/2 + k\pi, k$ an integer).
   * The hyperbolic co-tangent is not continuous at $x=0$ – when $\sinh$ is $0$,
-  * The semicircle $f(x) = \sqrt{1 - x^2}$ is *continuous* on $(-1, 1)$. It is not continuous at $-1$ and $1$, though it is right continuous at $-1$ and left continuous at $1$.
+  * The semicircle $f(x) = \sqrt{1 - x^2}$ is *continuous* on $(-1, 1)$. It is not continuous at $-1$ and $1$, though it is right continuous at $-1$ and left continuous at $1$. (It is continuous on $[-1,1]$.)
 
 
 ##### Examples of discontinuity
@@ -210,6 +212,13 @@ solve(del, c)
 This gives the value of $c$.
 
 
+This is a bit fussier than need be. As the left and right pieces (say, $f_l$ and $f_r$) as both are polynomials are continuous everywhere, so would have left and right limits given through evaluation. Solving for `c` as follows is enough:
+
+```{julia}
+solve(ex1(x=>0) ~ ex2(x=>0), c)
+```
+
+
 ## Rules for continuity
 
 
@@ -384,7 +393,7 @@ numericq(val)
 ###### Question
 
 
-Suppose $f(x)$, $g(x)$, and $h(x)$ are continuous functions on $(a,b)$. If $a < c < b$, are you sure that $lim_{x \rightarrow c} f(g(x))$ is $f(g(c))$?
+Suppose $f(x)$, $g(x)$, and $h(x)$ are continuous functions on $(a,b)$. If $a < c < b$, are you sure that $\lim_{x \rightarrow c} f(g(x))$ is $f(g(c))$?
 
 
 ```{julia}
@@ -453,7 +462,7 @@ yesnoq(true)
 ###### Question
 
 
-Let $f(x)$ and $g(x)$ be continuous functions whose graph of $[0,1]$ is given by:
+Let $f(x)$ and $g(x)$ be continuous functions. Their graphs over $[0,1]$ are given by:
 
 
 ```{julia}

quarto/limits/intermediate_value_theorem.qmd

@@ -76,7 +76,7 @@ ImageFile(imgfile, caption)
 In the early years of calculus, the intermediate value theorem was intricately connected with the definition of continuity, now it is a consequence.
 
 
-The basic proof starts with a set of points in $[a,b]$: $C = \{x \text{ in } [a,b] \text{ with } f(x) \leq y\}$. The set is not empty (as $a$ is in $C$) so it *must* have a largest value, call it $c$ (this requires the completeness property of the real numbers).  By continuity of $f$, it can be shown that $\lim_{x \rightarrow c-} f(x) = f(c) \leq y$ and $\lim_{y \rightarrow c+}f(x) =f(c) \geq y$, which forces $f(c) = y$.
+The basic proof starts with a set of points in $[a,b]$: $C = \{x \text{ in } [a,b] \text{ with } f(x) \leq y\}$. The set is not empty (as $a$ is in $C$) so it *must* have a largest value, call it $c$ (this might seem obvious, but it requires the completeness property of the real numbers).  By continuity of $f$, it can be shown that $\lim_{x \rightarrow c-} f(x) = f(c) \leq y$ and $\lim_{y \rightarrow c+}f(x) =f(c) \geq y$, which forces $f(c) = y$.
 
 
 ### Bolzano and the bisection method
@@ -87,8 +87,19 @@ Suppose we have a continuous function $f(x)$ on $[a,b]$ with $f(a) < 0$ and $f(b
 
 We use this fact when building a "sign chart" of a polynomial function. Between any two consecutive real zeros the polynomial can not change sign. (Why?) So a "test point" can be used to determine the sign of the function over an entire interval.
 
+The `sign_chart` function from `CalculusWithJulia` uses this to indicate where an *assumed* continuous function changes sign:
 
-Here, we use the Bolzano theorem to give an algorithm - the *bisection method* - to locate the value $c$ under the assumption $f$ is continuous on $[a,b]$ and changes sign between $a$ and $b$.
+```{julia}
+f(x) = sin(x + x^2) + x/2
+sign_chart(f, -3, 3)
+```
+
+
+The intermediate value theorem can find the sign of the function *between* adjacent zeros, but how are the zeros identified?
+
+Here, we use the Bolzano theorem to give an algorithm - the *bisection method* - to locate a value $c$ in $[a,b]$ with $f(c) = 0$ under the assumptions:
+* $f$ is continuous on $[a,b]$
+* $f$ changes sign between $a$ and $b$. (In particular, when  $f(a)$ and $f(b)$ have different signs.)
 
 ::: {.callout-note}
 #### Between
@@ -339,6 +350,13 @@ find_zero(h, (0, 2))
 ### Solving `f(x) = g(x)` and `f(x) = c`
 
 The above shows a means to translate a given problem into one that can be solved with `find_zero`. Basically to solve either when a function is a non-zero constant or when a function is equal to some other function, the difference between the two sides is formed and turned into a function, called `h` above.
+
+If using symbolic expressions, as below, then an equation (formed by `~`) can be passed to `find_zero`:
+
+```{julia}
+@syms x
+solve(cos(x) ~ x, (0, 2))
+```
 :::
 
 ##### Example: Inverse functions
@@ -493,7 +511,7 @@ Note that the function is infinite at `b`:
 d(b)
 ```
 
-From the graph,  we can see the zero is around `b`. As `d(b)` is `-Inf` we can use the bracket `(b/2,b)`
+From the graph,  we can see the zero is around `b`. As `d(b)` is `-Inf` we can use the bracket `(b/2, b)`
 
 
 ```{julia}
@@ -569,7 +587,7 @@ find_zero(f, I), find_zero(f, I, p=2)
 
 The second number is the solution when `p=2`.
 
-The above used a *keyword* argument, but a positional argument allows for broadcasting:
+The above used a *keyword* argument to pass in the parameter, but using a positional argument (the last one) allows for broadcasting:
 
 ```{julia}
 find_zero.(f, Ref(I), 1:5) # solutions for p=1,2,3,4,5

quarto/limits/limits.qmd

@@ -238,13 +238,13 @@ annotate!([(0,0+Δ,"A"), (x-Δ,y+Δ/4, "B"), (1+Δ/2,y/x, "C"),
 annotate!([(.2*cos(θ/2), 0.2*sin(θ/2), "θ")])
 imgfile = tempname() * ".png"
 savefig(p, imgfile)
-caption = "Triangle ``ABD` has less area than the shaded wedge, which has less area than triangle ``ACD``. Their respective areas are ``(1/2)\\sin(\\theta)``, ``(1/2)\\theta``, and ``(1/2)\\tan(\\theta)``. The inequality used to show ``\\sin(x)/x`` is bounded below by ``\\cos(x)`` and above by ``1`` comes from a division by ``(1/2) \\sin(x)`` and taking reciprocals.
+caption = "Triangle ``ABD`` has less area than the shaded wedge, which has less area than triangle ``ACD``. Their respective areas are ``(1/2)\\sin(\\theta)``, ``(1/2)\\theta``, and ``(1/2)\\tan(\\theta)``. The inequality used to show ``\\sin(x)/x`` is bounded below by ``\\cos(x)`` and above by ``1`` comes from a division by ``(1/2) \\sin(x)`` and taking reciprocals.
 "
 plotly()
 ImageFile(imgfile, caption)
 ```
 
-To discuss the case of $(1+x)^{1/x}$ it proved convenient to assume $x = 1/m$ for integer values of $m$. At the time of Cauchy, log tables were available to identify the approximate value of the limit. Cauchy computed the following value from logarithm tables.
+To discuss the case of $(1+x)^{1/x}$ it proved convenient to assume $x = 1/m$ for integer values of $m$. At the time of Cauchy, log tables were available to identify the approximate value of the limit. Cauchy computed the following value from logarithm tables:
 
 
 ```{julia}
@@ -337,13 +337,14 @@ Let's return to the function $f(x) = \sin(x)/x$. This function was studied by Eu
 
 ```{julia}
 #| hold: true
+
 f(x) = sin(x)/x
-xs, ys = unzip(f, -pi/2, pi/2)  # get points used to plot `f`
-plot(xs,  ys)
-scatter!(xs, ys)
+plot(f, -pi/2, pi/2;
+     seriestype=[:scatter, :line], # show points and line segments
+     legend=false)
 ```
 
-The $y$ values of the graph seem to go to $1$ as the $x$ values get close to $0$. (That the graph looks defined at $0$ is due to the fact that the points sampled to graph do not include $0$, as shown through the `scatter!` command – which can be checked via `minimum(abs, xs)`.)
+The $y$ values of the graph seem to go to $1$ as the $x$ values get close to $0$. (That the graph looks defined at $0$ is due to the fact that the points sampled to graph do not include $0$.)
 
 
 We can also verify Euler's intuition through this graph:
@@ -432,7 +433,7 @@ The `NaN` indicates that this function is indeterminate at $c=2$. A quick plot g
 ```{julia}
 #| hold: true
 c, delta = 2, 1
-plot(x -> (x^2 - 5x + 6) / (x^2 + x - 6), c - delta, c + delta)
+plot(f, c - delta, c + delta)
 ```
 
 The graph looks "continuous." In fact, the value $c=2$ is termed a *removable singularity* as redefining $f(x)$ to be $-0.2$ when $x=2$ results in a "continuous" function.
@@ -585,13 +586,11 @@ g(x) = (1 - cos(x)) / x^2
 g(0)
 ```
 
-What is the value of $L$, if it exists? A quick attempt numerically yields:
+What is the value of $L$, if it exists? Getting closer to $0$ numerically than the default yields:
 
 
 ```{julia}
-xs = 0 .+ hs
-ys = [g(x) for x in xs]
-[xs ys]
+lim(g, 0; n=9)
 ```
 
 Hmm, the values in `ys` appear to be going to $0.5$, but then end up at $0$. Is the limit $0$ or $1/2$? The answer is $1/2$. The last $0$ is an artifact of floating point arithmetic and the last few deviations from `0.5` due to loss of precision in subtraction. To investigate, we look more carefully at the two ratios:
@@ -606,7 +605,7 @@ y2s = [x^2 for x in xs]
 Looking at the bottom of the second column reveals the error. The value of `1 - cos(1.0e-8)` is `0` and not a value around `5e-17`, as would be expected from the pattern above it. This is because the smallest floating point value less than `1.0` is more than `5e-17` units away, so `cos(1e-8)` is evaluated to be `1.0`. There just isn't enough granularity to get this close to $0$.
 
 
-Not that we needed to. The answer would have been clear if we had stopped with `x=1e-6`, say.
+Not that we needed to. The answer would have been clear if we had stopped with `x=1e-6` (with `n=6`) say.
 
 
 In general, some functions will frustrate the numeric approach. It is best to be wary of results. At a minimum they should confirm what a quick graph shows, though even that isn't enough, as this next example shows.
@@ -633,23 +632,20 @@ A plot shows the answer appears to be straightforward:
 
 
 ```{julia}
-#| echo: false
 h(x) = x^2 + 1 + log(abs(11*x - 15))/99
 plot(h, 15/11 - 1, 15/11 + 1)
 ```
 
-Taking values near $15/11$ shows nothing too unusual:
+Taking values near $15/11$ shows nothing perhaps too unusual:
 
 
 ```{julia}
 #| hold: true
 c = 15/11
-hs = [1/10^i for i in 4:3:16]
-xs = c .+ hs
-[xs h.(xs)]
+lim(h, c; n = 16)
 ```
 
-(Though both the graph and the table hint at something a bit odd.)
+(Though the graph and table do hint at something a bit odd -- the graph shows a blip, the table doesn't show values in the second column going towards a specific value.)
 
 
 However the limit in this case is $-\infty$ (or DNE), as there is an aysmptote at $c=15/11$. The problem is the asymptote due to the logarithm is extremely narrow and happens between floating point values to the left and right of $15/11$.
@@ -726,7 +722,7 @@ For example, the limit at $0$ of $(1-\cos(x))/x^2$ is easily handled:
 limit((1 - cos(x)) / x^2, x => 0)
 ```
 
-The pair notation (`x => 0`) is used to indicate the variable and the value it is going to.
+The pair notation (`x => 0`) is used to indicate the variable and the value it is going to. A `dir` argument is used to indicate ``x \rightarrow c+`` (the default), ``x \rightarrow c-``, and ``x \rightarrow c``.
 
 
 ##### Example
@@ -736,16 +732,16 @@ We look again at this function which despite having a vertical asymptote at $x=1
 
 
 $$
-f(x) = x^2 + 1 + \log(| 11 \cdot x - 15 |)/99.
+h(x) = x^2 + 1 + \log(| 11 \cdot x - 15 |)/99.
 $$
 
 We find the limit symbolically at $c=15/11$ as follows, taking care to use the exact value `15//11` and not the *floating point* approximation returned by `15/11`:
 
 
 ```{julia}
 #| hold: true
-f(x) = x^2 + 1 + log(abs(11x - 15))/99
-limit(f(x), x => 15 // 11)
+h(x) = x^2 + 1 + log(abs(11x - 15))/99
+limit(h(x), x => 15 // 11)
 ```
 
 ##### Example
@@ -764,24 +760,26 @@ We have for the first:
 
 
 ```{julia}
-limit( (2sin(x) - sin(2x)) / (x - sin(x)), x => 0)
+limit( (2sin(x) - sin(2x)) / (x - sin(x)), x => 0; dir="+-")
 ```
 
+(The `dir = "+-"` indicates take both a right and left limit and ensure both exist and are equal.)
+
 The second is similarly done, though here we define a function for variety:
 
 
 ```{julia}
 #| hold: true
 f(x) = (exp(x) - 1 - x) / x^2
-limit(f(x), x => 0)
+limit(f(x), x => 0; dir="+-")
 ```
 
 Finally, for the third we define a new variable and proceed:
 
 
 ```{julia}
 @syms rho::real
-limit( (x^(1-rho) - 1) / (1 - rho), rho => 1)
+limit( (x^(1-rho) - 1) / (1 - rho), rho => 1; dir="+-")
 ```
 
 This last limit demonstrates that the `limit` function of `SymPy` can readily evaluate limits that involve parameters, though at times some assumptions on the parameters may be needed, as was done through `rho::real`.
@@ -842,7 +840,7 @@ limit(f(x), x => PI/2)
 Right and left limits will be discussed in the next section; here we give an example of the idea.  The mathematical convention is to say a limit exists if both the left *and* right limits exist and are equal. Informally a right (left) limit at $c$ only considers values of $x$ more (less) than $c$. The `limit` function of `SymPy`  finds directional limits by default, a right limit, where $x > c$.
 
 
-The left limit can be found by passing the argument `dir="-"`. Passing `dir="+-"` (and not `"-+"`) will compute the mathematical limit, throwing an error in `Python` if no limit exists.
+The left limit can be found by passing the argument `dir="-"`. Passing `dir="+-"` (and not `"-+"`), as done in a few examples above, will compute the mathematical limit, throwing an error in `Python` if no limit exists.
 
 
 ```{julia}
@@ -938,7 +936,7 @@ as this is the limit of $f(g(x))$ with $f$ as above and  $g(x) = x^2$. We need $
 ##### Example: products
 
 
-Consider this complicated limit found on this [Wikipedia](http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule) page.
+Consider this more complicated limit found on this [Wikipedia](http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule) page.
 
 
 $$