daviddoji/CalculusWithJuliaNotes.jl
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use quarto, not Pluto to render pages

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jverzani
2022-07-24 16:38:24 -04:00
parent 93c993206a
commit 7b37ca828c
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CwJ/precalc/transformations.jmd
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@@ -114,17 +114,13 @@ plot!(gf, label="g∘f")
```
```julia;echo=false
note("""
!!! note
Unlike how the basic arithmetic operations are treated, `Julia` defines the infix
Unicode operator `\\circ[tab]` to represent composition of functions,
mirroring mathematical notation. This infix operations takes in two functions and returns an anonymous function. It
can be useful and will mirror standard mathematical usage up to issues
with precedence rules.
Unlike how the basic arithmetic operations are treated, `Julia` defines the infix
Unicode operator `\\circ[tab]` to represent composition of functions,
mirroring mathematical notation. This infix operations takes in two functions and returns an anonymous function. It
can be useful and will mirror standard mathematical usage up to issues
with precedence rules.
""")
```
Starting with two functions and composing them requires nothing more
than a solid grasp of knowing the rules of function evaluation. If
@@ -163,11 +159,9 @@ other compositions could have been given above. For example, the last
function is also $f(x) = e^{-x/2}$ composed with $g(x) = x^2$.
```julia;echo=false
note("""
The real value of composition is to break down more complicated things into a sequence of easier steps. This is good mathematics, but also good practice more generally. For example, when we approach a problem with the computer, we generally use a smallish set of functions and piece them together (that is, compose them) to find a solution.
""")
```
!!! note
The real value of composition is to break down more complicated things into a sequence of easier steps. This is good mathematics, but also good practice more generally. For example, when we approach a problem with the computer, we generally use a smallish set of functions and piece them together (that is, compose them) to find a solution.
### Shifting and scaling graphs
@@ -508,8 +502,8 @@ If $f(x) = 1/x$ and $g(x) = x-2$, what is $g(f(x))$?
```julia; hold=true;echo=false
choices=["``1/(x-2)``", "``1/x - 2``", "``x - 2``", "``-2``"]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
###### Question
@@ -519,8 +513,8 @@ If $f(x) = e^{-x}$ and $g(x) = x^2$ and $h(x) = x-3$, what is $f \circ g \circ h
```julia; hold=true;echo=false
choices=["``e^{-x^2 - 3}``", "``(e^x -3)^2``",
"``e^{-(x-3)^2}``", "``e^x+x^2+x-3``"]
ans = 3
radioq(choices, ans)
answ = 3
radioq(choices, answ)
```
###### Question
@@ -530,8 +524,8 @@ If $h(x) = (f \circ g)(x) = \sin^2(x)$ which is a possibility for $f$ and $g$:
choices = [raw"``f(x)=x^2; \quad g(x) = \sin^2(x)``",
raw"```f(x)=x^2; \quad g(x) = \sin(x)``",
raw"``f(x)=\sin(x); \quad g(x) = x^2``"]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
@@ -544,7 +538,7 @@ choices = [
raw"``h(x) = 6 + \sin(x + 4)``",
raw"``h(x) = 6 + \sin(x-4)``",
raw"``h(x) = 6\sin(x-4)``"]
ans = 3
answ = 3
radioq(choices, 3)
```
@@ -555,8 +549,8 @@ Let $h(x) = 4x^2$ and $f(x) = x^2$. Which is **not** true:
choices = [L"The graph of $h(x)$ is the graph of $f(x)$ stretched by a factor of ``4``",
L"The graph of $h(x)$ is the graph of $f(x)$ scaled by a factor of ``2``",
L"The graph of $h(x)$ is the graph of $f(x) shifted up by ``4`` units"]
ans = 3
radioq(choices, ans)
answ = 3
radioq(choices, answ)
```
###### Question
@@ -567,8 +561,8 @@ The transformation $h(x) = (1/a) \cdot f((x-b)/a)$ can be viewed in one sequence
choices = [L"scaling by $1/a$, then shifting by $b$, then stretching by $1/a$",
L"shifting by $a$, then scaling by $b$, and then scaling by $1/a$",
L"shifting by $a$, then scaling by $a$, and then scaling by $b$" ]
ans=1
radioq(choices, ans)
answ=1
radioq(choices, answ)
```
###### Question
@@ -603,8 +597,8 @@ raw"``\sin(2x)``",
raw"``\sin(\pi x)``",
raw"``2 \sin(\pi x)``"
]
ans = 4
radioq(choices, ans)
answ = 4
radioq(choices, answ)
```
@@ -626,8 +620,8 @@ choices = [
q"D(S(f))(n) = f(n)",
q"S(D(f))(n) = f(n) - f(0)"
]
ans = 2
radioq(choices, ans, keep_order=true)
answ = 2
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -645,6 +639,6 @@ choices = [
q"D(S(f))(n) = f(n)",
q"S(D(f))(n) = f(n) - f(0)"
]
ans = 1
radioq(choices, ans, keep_order=true)
answ = 1
radioq(choices, answ, keep_order=true)
```