{
"cells": [
{
"cell_type": "markdown",
"id": "18ab9387-60ed-49dd-b641-b631c6673b53",
"metadata": {},
"source": [
"1. A pig gains weight according to a logistics curve such that the\n",
"weight after $t$ days is\n",
"$$\n",
" w(t)={800\\over 1+3e^{-t/30}}\\,\\hbox{lbs}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "8aa26c6e-1f83-4ef0-84a1-a118eb038a9d",
"metadata": {},
"source": [
"(i) Show that the pig is gaining about $5$ lbs/day at $t = 0$. What happens\n",
"as $t$ increases?"
]
},
{
"cell_type": "markdown",
"id": "e785cc2a-b64e-40fe-9623-ba9893c84c7f",
"metadata": {},
"source": [
"To show $w'(0)=5$ use the Symbolics library in Julia."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "2df92fcc-2277-4ff4-8abe-07d16e4582d2",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Vector{Num}:\n",
" t"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ENV[\"JULIA_COLORS\"] = false\n",
"using Symbolics\n",
"D(f,x)=expand_derivatives(Differential(x)(f))\n",
"@variables t"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "1d4f07c4-6403-4214-a8c7-7386bde0506a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[96m800\u001b[39m / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Define the function and check it is correct\n",
"w(t)=M/(1+3*exp(-t/30))\n",
"M=800\n",
"w(t)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "de72ecd9-35cf-4c19-b0fb-33692b8e87f1",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(\u001b[96m(\u001b[39m\u001b[96m80//1\u001b[39m\u001b[96m)\u001b[39m*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m)"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Create the derivative dw(t)\n",
"dwtmp=D(w(t),t)\n",
"dws=\"dw(t)=\"*string(dwtmp)\n",
"eval(Meta.parse(dws))\n",
"dw(t)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "81559b14-21e0-4715-9a0a-bfab8f220f75",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.0"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Evaluate dw(0)\n",
"dw(0)"
]
},
{
"cell_type": "markdown",
"id": "99ed54be-737e-4464-9cc1-35697f72fd37",
"metadata": {},
"source": [
"(ii) Suppose it costs $45$ cents a day to keep the pig and\n",
"the market price for pigs is $65$ cents per pound, but is\n",
"falling 1 cent per day. Find the optimal time to sell the pig."
]
},
{
"cell_type": "markdown",
"id": "3fe650e2-c7eb-4984-a94e-33140b7740f6",
"metadata": {},
"source": [
"Define the profit as\n",
"$$\n",
" P(t)=p(t)\\cdot w(t)-c(t)\n",
"$$\n",
"where $p(t)$ is the price of the pig at time $t$ and $c(t)$ is the cost of keeping the pig for $t$ days.\n",
"\n",
"Then solve $P'(t)=0$ to find the optimal time to sell the pig."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "c3a42aac-02aa-4428-927e-1973336e38b0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-\u001b[96m45\u001b[39mt + (\u001b[96m800\u001b[39m(\u001b[96m65\u001b[39m - t)) / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Create the profit function P(t) and display it\n",
"p(t)=65-t\n",
"c(t)=45*t\n",
"P(t)=p(t)*w(t)-c(t)\n",
"P(t)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "4a676a05-350c-47f2-816c-51d83922fb1e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-\u001b[96m45\u001b[39m + (\u001b[96m(\u001b[39m\u001b[96m80//1\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + \u001b[96m-800\u001b[39m / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Create the derivative dP(t)\n",
"dPtmp=D(P(t),t)\n",
"dPs=\"dP(t)=\"*string(dPtmp)\n",
"eval(Meta.parse(dPs))\n",
"dP(t)"
]
},
{
"cell_type": "markdown",
"id": "f2b878fc-2ab1-4f3c-8778-206be4eb2b3e",
"metadata": {},
"source": [
"Since the $P'(t)$ is nonlinear use Newton's method to approximate a solution. First graph $P(t)$ to obtain a good initial guess of the maximum and then iterate\n",
"$$\n",
" t_{n+1}=t_n-g(t)/g'(t)\n",
"$$\n",
"where $g(t)=P'(t)$ and $g'(t)=P''(t)$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "79ac27b0-eecf-4000-8c3e-19d73e69c080",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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",
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\")
"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Plots\n",
"plot(P,0:0.1:20,label=\"P(t)\")"
]
},
{
"cell_type": "markdown",
"id": "e28ac470-17d3-44d8-8f83-9deba872ee45",
"metadata": {},
"source": [
"From the graph it appears the $t_0=12$ is a reasonable initial guess for Newton's method."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "a2f831f5-7108-4750-b152-baa342c5928d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-\u001b[96m(\u001b[39m\u001b[96m80//1\u001b[39m\u001b[96m)\u001b[39m*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t) - \u001b[96m(\u001b[39m\u001b[96m8//3\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + (\u001b[96m(\u001b[39m\u001b[96m16//1\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*(exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)^\u001b[96m2\u001b[39m)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m3\u001b[39m) + (\u001b[96m(\u001b[39m\u001b[96m-80//1\u001b[39m\u001b[96m)\u001b[39m*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g(t)=dP(t)\n",
"# Create the derivative dg(t)\n",
"dgtmp=D(g(t),t)\n",
"dgs=\"dg(t)=\"*string(dgtmp)\n",
"eval(Meta.parse(dgs))\n",
"dg(t)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "d97991ac-7bbf-4e81-b2ae-d75668c182d3",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"t_1=12.337004792474968\n",
"t_2=12.334891525596113\n",
"t_3=12.334891442632154\n",
"t_4=12.334891442632161\n",
"t_5=12.334891442632161\n",
"\n",
"The optimal time to sell the pig is 12.334891442632161\n",
"The optimal profit is 13542.357422246736\n"
]
}
],
"source": [
"# Perform Newton's iterations\n",
"tn=12.0\n",
"for n=1:5\n",
" tn=tn-g(tn)/dg(tn)\n",
" println(\"t_$n=\",tn)\n",
"end\n",
"topt=tn\n",
"Popt=P(topt)\n",
"println(\"\\nThe optimal time to sell the pig is \",topt)\n",
"println(\"The optimal profit is \",Popt)"
]
},
{
"cell_type": "markdown",
"id": "485181b5-0b92-4ebc-98a1-195a3528a78b",
"metadata": {},
"source": [
"(iii) The parameter $800$ represents the eventual mature weight $M$\n",
"of the pig. Thus,\n",
"$$\n",
" w(t)={M\\over 1+3e^{-t/30}}\\,\\hbox{lbs}.\n",
"$$\n",
"Perform a sensitivity analysis for the parameter $M$. Compute\n",
"$S(t,M)$ and $S(P,M)$ evaluated at $M=800$. Here $P$ is the\n",
"profit obtained at the optimal time $t$ to sell the pig."
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "5ce11144-87d3-4d7b-bc91-0b9b33bd2d3b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Vector{Num}:\n",
" M"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Change M to a variable\n",
"@variables M"
]
},
{
"cell_type": "markdown",
"id": "dbe2152f-394a-4f16-b478-d06c261197c8",
"metadata": {},
"source": [
"Note the above idea to change $M$ to a variable won't work in Pluto\n",
"because in order to keep track of dependencies Pluto doesn't allow\n",
"one to change the definition of a variable in the middle of a calculation.\n",
"For Pluto, one would need to start over with a new function $\\widetilde{\\it w}(t)$ and\n",
"a different name for the variable $M$. Here we are using JupyterLab which\n",
"works the same as at the Julia command line so we can simply redefine $M$."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "da34780a-dcf0-45b8-bdee-a11ba3501902",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"M / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Verify p(t) not has M in it\n",
"w(t)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "c61f6460-48e7-4343-a077-569eac8a97ea",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-\u001b[96m45\u001b[39m + (\u001b[96m(\u001b[39m\u001b[96m1//10\u001b[39m\u001b[96m)\u001b[39m*M*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + (-M) / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Recompute dP(t) and g(t)\n",
"dPtmp=D(P(t),t)\n",
"dPs=\"dP(t)=\"*string(dPtmp)\n",
"eval(Meta.parse(dPs))\n",
"g(t)=dP(t)\n",
"g(t)"
]
},
{
"cell_type": "markdown",
"id": "df389c1f-63ab-42d5-919a-27a9622cdd27",
"metadata": {},
"source": [
"To compute $dt/dM$ use implicit differentiation and the fact that $g(t)=0$.\n",
"Thus,\n",
"$$\n",
" {dg\\over ds}={\\partial g\\over\\partial t}{dt\\over dM}+{\\partial g\\over\\partial M}=0\n",
"$$\n",
"implies\n",
"$$\n",
" {dt\\over dM}=-{\\partial g\\over\\partial M}\\Big/{\\partial g\\over\\partial t}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "51106637-2b17-4b92-8609-32ac3e744097",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(\u001b[96m(\u001b[39m\u001b[96m1//10\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + \u001b[96m-1\u001b[39m / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dgdM=D(g(t),M)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "d5c58580-eac1-4e23-98d1-6dc3f3668298",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(\u001b[96m(\u001b[39m\u001b[96m1//50\u001b[39m\u001b[96m)\u001b[39m*M*(\u001b[96m65\u001b[39m - t)*(exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)^\u001b[96m2\u001b[39m)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m3\u001b[39m) + (\u001b[96m(\u001b[39m\u001b[96m-1//10\u001b[39m\u001b[96m)\u001b[39m*M*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + (-\u001b[96m(\u001b[39m\u001b[96m1//10\u001b[39m\u001b[96m)\u001b[39m*M*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t) - \u001b[96m(\u001b[39m\u001b[96m1//300\u001b[39m\u001b[96m)\u001b[39m*M*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m)"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dgdt=D(g(t),t)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "b8260a52-5306-494f-94c8-972763b6b938",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"S(t,M)=0.4329428182079427\n"
]
}
],
"source": [
"# Now evaluate S(t,M) at t=tn and M=800\n",
"StMtmp=substitute(-M/t*dgdM/dgdt,[t=>topt,M=>800])\n",
"StM=eval(Symbolics.toexpr(StMtmp))\n",
"println(\"S(t,M)=\",StM)"
]
},
{
"cell_type": "markdown",
"id": "8d0ee757-0fa2-4924-93ec-785a3c2b6db0",
"metadata": {},
"source": [
"To compute $S(P,M)$ note that\n",
"$$\n",
" {dP\\over dM}={\\partial P\\over\\partial t}{dt\\over dM}+{\\partial P\\over\\partial M}\n",
"$$\n",
"and at the optimal time $t$ that $\\partial P/\\partial t=0$. Therefore\n",
"$$\n",
" S(P,M)={M\\over P} {\\partial P\\over\\partial M}\\bigg|_{t=t_n,M=800}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "89af0416-663b-4e83-98b5-d355a1452213",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-\u001b[96m45\u001b[39mt + (M*(\u001b[96m65\u001b[39m - t)) / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P(t)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "7585aecd-f4e7-4647-9b1e-3ea3e908404c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(\u001b[96m65\u001b[39m - t) / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dPdM=D(P(t),M)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "7d5fef06-9b1e-4837-9239-f63780b80916",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"17.621784421456482"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eval(Symbolics.toexpr(substitute(dPdM,[t=>topt,M=>800])))"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "f268cc7a-884e-45ca-ba77-e79605cb56d0",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"S(P,M)=1.0409877023336132\n"
]
}
],
"source": [
"SPMtmp=substitute(M/P(t)*dPdM,[t=>topt,M=>800])\n",
"SPM=eval(Symbolics.toexpr(SPMtmp))\n",
"println(\"S(P,M)=\",SPM)"
]
},
{
"cell_type": "markdown",
"id": "d9c93143-813f-48e4-b008-179ee75374e2",
"metadata": {},
"source": [
"Note that it is also possible to approximate the sensitivity by considering $M=801$ and\n",
"recomputing the optimal time and then using\n",
"$$\n",
" {dt\\over dM}\\approx{\\Delta t\\over\\Delta M}\n",
"\\qquad\\hbox{and}\\qquad\n",
" {dP\\over dM}\\approx{\\Delta P\\over\\Delta M}\n",
"$$\n",
"We do that now as an optional check on the above answer."
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "b208973d-870c-4e00-adfa-6c19f4b10774",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-\u001b[96m45\u001b[39m + (\u001b[96m(\u001b[39m\u001b[96m801//10\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + \u001b[96m-801\u001b[39m / (\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Optional way of approximating S(t,M) and S(P,M)\n",
"M=801\n",
"dPtmp=D(P(t),t)\n",
"dPs=\"dP(t)=\"*string(dPtmp)\n",
"eval(Meta.parse(dPs))\n",
"g(t)=dP(t)\n",
"g(t)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "daf14df8-730b-41de-827d-8ac15f4be325",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(\u001b[96m(\u001b[39m\u001b[96m801//50\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*(exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)^\u001b[96m2\u001b[39m)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m3\u001b[39m) + (-\u001b[96m(\u001b[39m\u001b[96m801//10\u001b[39m\u001b[96m)\u001b[39m*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t) - \u001b[96m(\u001b[39m\u001b[96m267//100\u001b[39m\u001b[96m)\u001b[39m*(\u001b[96m65\u001b[39m - t)*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m) + (\u001b[96m(\u001b[39m\u001b[96m-801//10\u001b[39m\u001b[96m)\u001b[39m*exp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t)) / ((\u001b[96m1\u001b[39m + \u001b[96m3\u001b[39mexp(\u001b[96m(\u001b[39m\u001b[96m-1//30\u001b[39m\u001b[96m)\u001b[39m*t))^\u001b[96m2\u001b[39m)"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dgtmp=D(g(t),t)\n",
"dgs=\"dg(t)=\"*string(dgtmp)\n",
"eval(Meta.parse(dgs))\n",
"dg(t)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "06f367d2-ddb2-4f7e-9547-a79afc3a2d02",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"t_1=12.34375594742852\n",
"t_2=12.341557751357138\n",
"t_3=12.341557661617747\n",
"t_4=12.341557661617747\n",
"t_5=12.341557661617747\n",
"\n",
"The new optimal time to sell the pig is 12.341557661617747\n"
]
}
],
"source": [
"# Perform Newton's iterations\n",
"tn=12.0\n",
"for n=1:5\n",
" tn=tn-g(tn)/dg(tn)\n",
" println(\"t_$n=\",tn)\n",
"end\n",
"tnew=tn\n",
"println(\"\\nThe new optimal time to sell the pig is \",tnew)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "aa7ebb47-2f22-469a-a78f-eb37e69b5682",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The approximate S(t,M)=0.43234877366143437\n"
]
}
],
"source": [
"StMapp=800/topt*(tnew-topt)/(M-800)\n",
"println(\"The approximate S(t,M)=\",StMapp)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "def40601-dcef-47f3-846e-7a504af5474b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The approximate S(P,M)=1.040998778405237\n"
]
}
],
"source": [
"SPMapp=800/Popt*(P(tnew)-Popt)/(M-800)\n",
"println(\"The approximate S(P,M)=\",SPMapp)"
]
},
{
"cell_type": "markdown",
"id": "1b0e74eb-972c-4380-821e-f8641822aa69",
"metadata": {},
"source": [
"Note that the approximate values for $S(t,M)$ and $S(P,M)$ computed using $M=801$\n",
"are close to the values obtained using implicit differentiation and the chain\n",
"rule in the earlier calculation. This consistency suggests both answers are correct."
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fe68103b-5e6b-42d4-aa64-2df60653c7d1",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.10",
"language": "julia",
"name": "julia-1.10"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.10.10"
}
},
"nbformat": 4,
"nbformat_minor": 5
}