daviddoji/CalculusWithJuliaNotes.jl
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2022-07-24 16:38:24 -04:00
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CwJ/derivatives/implicit_differentiation.jmd
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@@ -237,17 +237,12 @@ Not quite what we expect, perhaps, but substituting in ``f(x)/g(x)`` for ``y`` g
\frac{dy}{dx} = \frac{f'(x) - \frac{f(x)}{g(x)} g'(x)}{g(x)} = \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2}.
```
```julia; echo=false
note("""
In this example we mix notations using ``g'(x)`` to
represent a derivative of ``g`` with respect to ``x`` and ``dy/dx`` to
represent the derivative of ``y`` with respect to ``x``. This is done to
emphasize the value that we are solving for. It is just a convention
though, we could just as well have used the "prime" notation for each.
""")
```
!!! note
In this example we mix notations using ``g'(x)`` to
represent a derivative of ``g`` with respect to ``x`` and ``dy/dx`` to
represent the derivative of ``y`` with respect to ``x``. This is done to
emphasize the value that we are solving for. It is just a convention
though, we could just as well have used the "prime" notation for each.
##### Example: Graphing a tangent line
@@ -399,17 +394,11 @@ Basically this includes all the same steps as if done "by hand." Some effort cou
values for the parameters been substituted initially, but not doing so
shows their dependence in the derivative.
```julia; echo=false
alert("The use of `lambdify(H)` is needed to turn the symbolic expression, `H`, into a function.")
```
!!! warning
The use of `lambdify(H)` is needed to turn the symbolic expression, `H`, into a function.
```julia; echo=false
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.
""")
```
!!! 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.
## Higher order derivatives
@@ -818,8 +807,8 @@ choices = [
"``b \\cdot (1 - (x/a)^n)^{1/n}``",
"``-(x/a)^n / (y/b)^n``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -863,8 +852,8 @@ choices = [
"``2xy / (x^2 + a^2)``",
"``a^3/(x^2 + a^2)``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -885,8 +874,8 @@ Using Implicit differentiation, find when ``dy/dx = 0``.
```julia; hold=true; echo=false
choices = ["``y^2 = 3x/a``", "``y=3x^2/a``", "``y=a/(3x^2)``", "``y^2=a/(3x)``"]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
Substituting the correct value of ``y``, above, into the defining equation gives what value for ``x``:
@@ -898,8 +887,8 @@ choices=[
"``x=(1/2) a^3 3^{1/3}``",
"``x=(1/3) a^2 2^{1/2}``"
]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
###### Question
@@ -924,8 +913,8 @@ If ``y>0`` is the sign positive or negative?
```julia; hold=true; echo=false
choices = ["positive", "negative", "Can be both"]
ans = 2
radioq(choices, ans, keep_order=true)
answ = 2
radioq(choices, answ, keep_order=true)
```
If ``x>0`` is the sign positive or negative?
@@ -933,16 +922,16 @@ If ``x>0`` is the sign positive or negative?
```julia; hold=true; echo=false
choices = ["positive", "negative", "Can be both"]
ans = 3
radioq(choices, ans, keep_order=true)
answ = 3
radioq(choices, answ, keep_order=true)
```
When ``x>0``, the graph of the equation is...
```julia; hold=true; echo=false
choices = ["concave up", "concave down", "both concave up and down"]
ans = 3
radioq(choices, ans, keep_order=true)
answ = 3
radioq(choices, answ, keep_order=true)
```