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"source": [
"# Maximization and minimization with Julia\n"
]
},
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"source": [
"To get started, we load our `MTH229` package for plotting and other features:\n"
]
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"┌ Info: For saving to png with the Plotly backend PlotlyBase has to be installed.\n",
"└ @ Plots /Users/verzani/.julia/packages/Plots/1KWPG/src/backends.jl:318\n"
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"Plots.PlotlyBackend()"
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"source": [
"using MTH229\n",
"using Plots\n",
"plotly()"
]
},
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"source": [
"### Quick background\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Read about this material here: [Maximization and minimization with julia](http://mth229.github.io/extrema.html).\n"
]
},
{
"cell_type": "markdown",
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"source": [
"For the impatient, *extrema* is nothing more than a fancy word for describing either a maximum *or* a minimum. In calculus, we have **two** concepts of these: *relative* extrema and *absolute* extrema. Let's focus for a second on *absolute* extrema which are defined through:\n"
]
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{
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"> A value $y=f(x)$ is an *absolute maximum* over an interval $[a,b]$ if $y \\geq f(x)$ for all $x$ in $[a,b]$. (An absolute minimum has $y \\leq f(x)$ instead.)\n",
"\n"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"Of special note is that an absolute extrema involves *both* a function **and** an interval.\n"
]
},
{
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"source": [
"There are two theorems which help identify extrema here. The first, due to Bolzano, says that any continuous function on a *closed* interval will *necessarily* have an absolute maximum and minimum on that interval. The second, due to Fermat, tells us where to look: these absolute maximums and minimums can only occur at end points or critical points.\n"
]
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"source": [
"Bolzano and Fermat are historic figures. For us, we can plot a function to visually see extrema. The value of Bolzano is the knowledge that yes, plotting isn't a waste of time, as we are *guaranteed* to see what we look for. The value of Fermat is that if we want to get *precise* numeric answers, we have a means: identify the end points and the critical points then compare the function at *just* these values.\n"
]
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n"
]
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"source": [
"## An example\n"
]
},
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"source": [
"Among all rectangles with perimeter 24 find the one with maximum area.\n"
]
},
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"source": [
"This problem leads to two equations:\n"
]
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{
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" * a constraint based on a fixed perimeter: $25 = 2b + 2h$.\n",
" * an expression for the area: $A=h \\cdot b$.\n"
]
},
{
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"metadata": {},
"source": [
"We translate these into `Julia` functions. First, using the constraint, we solve for one variable and then substitute this in:\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"A (generic function with 1 method)"
]
},
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"output_type": "execute_result"
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],
"source": [
"h(b) = (25 - 2b) / 2\n",
"A(b) = h(b) * b # A = h * b translates to this"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So we have area as a function of a single variable. Here `b` ranges from $0$ to no more than 10, as both `b` and `h` need be non-negative. A plot shows the function to maximize:\n"
]
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"plot(A, 0, 10)"
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{
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"source": [
"Not only do we see a maximum value, we also can tell more:\n"
]
},
{
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"source": [
" * the maximum happens at a critical point - not an end point\n",
" * there is a unique critical point on this interval $[0,10]$.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"So, we can use `fzero` to find the critical point:\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
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"6.25"
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],
"source": [
"x = fzero(A', 5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We store the value as `x`. Is this the answer? Not quite, the question asks for the rectangle that gives the maximum area, so we should also find the height. But this is just\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
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"6.25"
]
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],
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"h(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In fact, for the problems encountered within, the critical point, the constraint at the critical point, or the function evaluated at the critical point are used to answer the questions:\n"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
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"(6.25, 6.25, 39.0625)"
]
},
"execution_count": 6,
"metadata": {},
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}
],
"source": [
"x, h(x), A(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Play this video to see this problem again and a related problem:\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[](https://asciinema.org/a/4tdlumFtE2mBswLZJSw6q9K9J)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"(This uses the Julia REPL, not a notebook; the begin/end block just combines commands like a cell.)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n"
]
},
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}
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