daviddoji/CalculusWithJuliaNotes.jl
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setup for *possible* CI rendering

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jverzani
2022-09-10 13:39:23 -04:00
parent decef62bbb
commit 2739cda456
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8
quarto/derivatives/implicit_differentiation.qmd
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@@ -9,7 +9,6 @@ This section uses these add-on packages:
```{julia}
using CalculusWithJulia
using Plots
using ImplicitPlots
using Roots
using SymPy
```
@@ -45,10 +44,10 @@ The **graph** of an equation is just the set of solutions to the equation repres
With this definition, the graph of a function $f(x)$ is just the graph of the equation $y = f(x)$. In general, graphing an equation is more complicated than graphing a function. For a function, we know for a given value of $x$ what the corresponding value of $f(x)$ is through evaluation of the function. For equations, we may have $0$, $1$ or more $y$ values for a given $x$ and even more problematic is we may have no rule to find these values.
There are a few options for plotting equations in `Julia`. We will use `ImplicitPlots` in this section, but note both `ImplicitEquations` and `IntervalConstraintProgramming` offer alternatives that are a bit more flexible.
There are a few options for plotting equations in `Julia`. We will use a function from the `ImplicitPlots` package that is in the `CalculusWithJulia` package in this section, but note both `ImplicitEquations` and `IntervalConstraintProgramming` offer alternatives that are a bit more flexible.
To plot an implicit equation using `ImplicitPlots` requires expressing the relationship in terms of a function, and then plotting the equation `f(x,y) = 0`. In practice this simply requires all the terms be moved to one side of an equals sign.
To plot an implicit equation using `implicit_plot` requires expressing the relationship in terms of a function, and then plotting the equation `f(x,y) = 0`. In practice this simply requires all the terms be moved to one side of an equals sign.
To plot the circle of radius $2$, or the equations $x^2 + y^2 = 2^2$ we would move all terms to one side $x^2 + y^2 - 2^2 = 0$ and then express the left hand side through a function:
@@ -397,7 +396,7 @@ The use of `lambdify(H)` is needed to turn the symbolic expression, `H`, into a
:::{.callout-note}
## Note
While `SymPy` itself has the `plot_implicit` function for plotting implicit equations, this works only with `PyPlot`, not `Plots`, so we use the `ImplicitPlots` package in these examples.
While `SymPy` itself has the `plot_implicit` function for plotting implicit equations, this works only with `PyPlot`, not `Plots`, so we use the `implicit_plot` function from the `ImplicitPlots` package in these examples.
:::
@@ -1081,4 +1080,3 @@ A plot can be made of either the boundary, the interior, or both.
#| eval: false
plot(region.inner) # plot interior; use r.boundary for boundary
```