daviddoji/CalculusWithJuliaNotes.jl
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jverzani
2022-07-25 20:51:08 -04:00
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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