use plotly; fix bitrot

quarto/derivatives/newtons_method.qmd

@@ -9,6 +9,7 @@ This section uses these add-on packages:
 ```{julia}
 using CalculusWithJulia
 using Plots
+plotly()
 using SymPy
 using Roots
 ```
@@ -20,7 +21,7 @@ using Roots
 The Babylonian method is an algorithm to find an approximate value for $\sqrt{k}$. It was described by the first-century Greek mathematician Hero of [Alexandria](http://en.wikipedia.org/wiki/Babylonian_method).
 
 
-The method starts with some initial guess, called $x_0$. It then applies a formula to produce an improved guess. This is repeated until the improved guess is accurate enough or it is clear the algorithm fails to work.
+The method starts with some initial guess, called $x_0$. This is usually some nearby value to the answer. The method then applies a formula to produce an improved guess. This is repeated until the improved guess is accurate enough or it is clear the algorithm fails to work.
 
 
 For the Babylonian method, the next guess, $x_{i+1}$, is derived from the current guess, $x_i$. In mathematical notation, this is the updating step:
@@ -250,7 +251,7 @@ nothing
 #| echo: false
 #| cache: true
 ### {{{newtons_method_example}}}
-
+gr()
 caption = """
 
 Illustration of Newton's Method converging to a zero of a function.
@@ -266,7 +267,7 @@ end
 
 imgfile = tempname() * ".gif"
 gif(anim, imgfile, fps = 1)
-
+plotly()
 ImageFile(imgfile, caption)
 ```
 
@@ -730,6 +731,7 @@ What can go wrong when one of these isn't the case is illustrated next:
 #| echo: false
 #| cache: true
 ### {{{newtons_method_poor_x0}}}
+gr()
 caption = """
 
 Illustration of Newton's Method converging to a zero of a function,
@@ -748,7 +750,7 @@ end
 
 imgfile = tempname() * ".gif"
 gif(anim, imgfile, fps = 2)
-
+plotly()
 ImageFile(imgfile, caption)
 ```
 
@@ -757,6 +759,7 @@ ImageFile(imgfile, caption)
 #| echo: false
 #| cache: true
 # {{{newtons_method_flat}}}
+gr()
 caption = L"""
 
 Illustration of Newton's method failing to converge as for some $x_i$,
@@ -777,7 +780,7 @@ anim = @animate for i=1:n
 end
 imgfile = tempname() * ".gif"
 gif(anim, imgfile, fps = 1)
-
+plotly()
 ImageFile(imgfile, caption)
 ```
 
@@ -789,7 +792,7 @@ ImageFile(imgfile, caption)
 #| echo: false
 #| cache: true
 # {{{newtons_method_cycle}}}
-
+gr()
 fn, a, b, c, = x -> abs(x)^(0.49),  -2, 2, 1.0
 caption = L"""
 
@@ -808,6 +811,7 @@ end
 imgfile = tempname() * ".gif"
 gif(anim, imgfile, fps = 2)
 
+plotly()
 ImageFile(imgfile, caption)
 ```
 
@@ -819,7 +823,7 @@ ImageFile(imgfile, caption)
 #| echo: false
 #| cache: true
 # {{{newtons_method_wilkinson}}}
-
+gr()
 caption = L"""
 
 The function $f(x) = x^{20} - 1$ has two bad behaviours for Newton's
@@ -845,7 +849,7 @@ anim = @animate for i=1:n
 end
 imgfile = tempname() * ".gif"
 gif(anim, imgfile, fps = 1)
-
+plotly()
 ImageFile(imgfile, caption)
 ```