daviddoji/CalculusWithJuliaNotes.jl
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some typos.

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This commit is contained in:
Fang Liu
2025-05-23 16:20:13 +08:00
parent 837a8eb42d
commit 4d0a9e9a72
10 changed files with 54 additions and 60 deletions
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24
quarto/derivatives/optimization.qmd
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@@ -639,21 +639,21 @@ It is hard to tell which would minimize time without more work. To check a case
```{julia}
x_straight = t(r1 =>2r0, b=>0, x=>out[1], a=>1, L=>2, r0 => 1) # for x=L
x_straight = subs(t, r1 =>2r0, b=>0, x=>out[1], a=>1, L=>2, r0 => 1) # for x=L
```
Compared to the smaller ($x=\sqrt{3}a/3$):
```{julia}
x_angle = t(r1 =>2r0, b=>0, x=>out[2], a=>1, L=>2, r0 => 1)
x_angle = subs(t, r1 =>2r0, b=>0, x=>out[2], a=>1, L=>2, r0 => 1)
```
What about $x=0$?
```{julia}
x_bent = t(r1 =>2r0, b=>0, x=>0, a=>1, L=>2, r0 => 1)
x_bent = subs(t, r1 =>2r0, b=>0, x=>0, a=>1, L=>2, r0 => 1)
```
The value of $x=\sqrt{3}a/3$ minimizes time:
@@ -671,7 +671,7 @@ Will this approach always be true? Consider different parameters, say we switch
```{julia}
pts = [0, out...]
m,i = findmin([t(r1 =>2r0, b=>0, x=>u, a=>2, L=>1, r0 => 1) for u in pts]) # min, index
m,i = findmin([subs(t, r1 =>2r0, b=>0, x=>u, a=>2, L=>1, r0 => 1) for u in pts]) # min, index
m, pts[i]
```
@@ -997,7 +997,7 @@ A rain gutter is constructed from a 30" wide sheet of tin by bending it into thi
2 * (1/2 * 10*cos(pi/4) * 10 * sin(pi/4)) + 10*sin(pi/4) * 10
```
Find a value in degrees that gives the maximum. (The first task is to write the area in terms of $\theta$.
Find a value in degrees that gives the maximum. (The first task is to write the area in terms of $\theta$.)
```{julia}
@@ -1049,7 +1049,7 @@ plot!(p, [0, 30,30,0], [0,10,30,0], color=:orange)
annotate!(p, [(x,y,l) for (x,y,l) in zip([15, 5, 31, 31], [1.5, 3.5, 5, 20], ["x=30", "θ", "10", "20"])])
```
What value of $x$ gives the largest angle $\theta$? (In degrees.)
What value of the largest angle $\theta$ that $x$ gives? (In degrees.)
```{julia}
@@ -1094,7 +1094,7 @@ radioq(choices, answ)
##### Question
Let $x_1$, $x_2$, $x_n$ be a set of unspecified numbers in a data set. Form the expression $s(x) = (x-x_1)^2 + \cdots (x-x_n)^2$. What is the smallest this can be (in $x$)?
Let $x_1$, $x_2$, $\dots, x_n$ be a set of unspecified numbers in a data set. Form the expression $s(x) = (x-x_1)^2 + \cdots + (x-x_n)^2$. What is the smallest this can be (in $x$)?
We approach this using `SymPy` and $n=10$
@@ -1108,7 +1108,7 @@ s(x) = sum((x-xi)^2 for xi in xs)
cps = solve(diff(s(x), x), x)
```
Run the above code. Baseed on the critical points found, what do you guess will be the minimum value in terms of the values $x_1$, $x_2, \dots$?
Run the above code. Based on the critical points found, what do you guess will be the minimum value in terms of the values $x_1$, $x_2, \dots$?
```{julia}
@@ -1117,7 +1117,7 @@ Run the above code. Baseed on the critical points found, what do you guess will
choices=[
"The mean, or average, of the values",
"The median, or middle number, of the values",
L"The square roots of the values squared, $(x_1^2 + \cdots x_n^2)^2$"
L"The square roots of the values squared, $(x_1^2 + \cdots + x_n^2)^2$"
]
answ = 1
radioq(choices, answ)
@@ -1126,7 +1126,7 @@ radioq(choices, answ)
###### Question
Minimize the function $f(x) = 2x + 3/x$ over $(0, \infty)$.
Find $x$ to minimize the function $f(x) = 2x + 3/x$ over $(0, \infty)$.
```{julia}
@@ -1190,7 +1190,7 @@ The width is:
w(h) = 12_000 / h
S(w, h) = (w- 2*8) * (h - 2*32)
S(h) = S(w(h), h)
hstar =find_zero(D(S), 500)
hstar = find_zero(D(S), 200)
wstar = w(hstar)
numericq(wstar)
```
@@ -1204,7 +1204,7 @@ The height is?
w(h) = 12_000 / h
S(w, h) = (w- 2*8) * (h - 2*32)
S(h) = S(w(h), h)
hstar =find_zero(D(S), 500)
hstar = find_zero(D(S), 200)
numericq(hstar)
```