### A Pluto.jl notebook ###
# v0.20.21

using Markdown
using InteractiveUtils

# ╔═╡ 73f73402-d472-442e-8ffe-1cb35fb979a1
using Symbolics

# ╔═╡ 72e83b1a-241c-42a0-aee5-7a7cb6fa0f10
md"""
Example 3.1. Reconsider the pig problem of Example 1.1, but now take into
account the fact that the growth rate of the pig is not constant. Assume that
the pig is young, so that the growth rate is increasing. When should we sell the
pig for maximum profit?
"""

# ╔═╡ 5574a419-825b-4b30-bd0b-4dcd7e858a28
md"""
Let ``w(t)`` be the weight of the pig at time ``t``.  Then
```math
	w(t)=w_0 e^{ct}
```
is the exponential rate of growth we consider here.
"""

# ╔═╡ f20ca194-53a3-4851-ad35-3bbdbb084ec9
C(t)=0.45*t

# ╔═╡ 0cf599f2-1213-4558-b27d-99ea3dd0920a
p(t)=0.65-0.01*t

# ╔═╡ 5e7e25dc-4466-4f0e-8803-a98d80387aae
D(f,x)=expand_derivatives(Differential(x)(f))

# ╔═╡ 15d2491c-90ad-4121-8535-48290e52755c
@variables t,c

# ╔═╡ bdad2f4a-b299-4c8d-9ef0-9d41ac7d3546
w(t)=200*exp(c*t)

# ╔═╡ 8bfd45ac-64e8-4967-a285-15b1ed8774f4
R(t)=w(t)*p(t)

# ╔═╡ 680ea528-57bf-4bc0-a8e9-bff97037f653
P(t)=R(t)-C(t)

# ╔═╡ 95245521-015c-4a21-936e-5ef8b8c0757c
dPdt=D(P(t),t)

# ╔═╡ d76788aa-45d7-4530-a00b-65cb6adba7b8
parudt=D(dPdt,t)

# ╔═╡ 9cd0cebf-d05b-4fd1-a569-9b290f1adf3c
parudc=expand(D(dPdt,c))

# ╔═╡ 7e1184e0-152a-4d4b-9d68-b0b8b428540d
dtdc=-parudc/parudt

# ╔═╡ d7b84b1c-6b93-47c4-930d-9dac9b8cdce0
Stc=substitute(c/t*dtdc,[c=>1/40,t=>19.46815944415122])

# ╔═╡ 9240ccb8-e05c-4ed3-99db-f0b5869e1ad7
md"""
The sensititivity is ``S(t,c)\approx 2.8746311019686357``.
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
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deps = ["SHA"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.ReadOnlyArrays]]
git-tree-sha1 = "e6f7ddf48cf141cb312b078ca21cb2d29d0dc11d"
uuid = "988b38a3-91fc-5605-94a2-ee2116b3bd83"
version = "0.2.0"

[[deps.RecipesBase]]
deps = ["PrecompileTools"]
git-tree-sha1 = "5c3d09cc4f31f5fc6af001c250bf1278733100ff"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.4"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "62389eeff14780bfe55195b7204c0d8738436d64"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.1"

[[deps.RuntimeGeneratedFunctions]]
deps = ["ExprTools", "SHA", "Serialization"]
git-tree-sha1 = "7257165d5477fd1025f7cb656019dcb6b0512c38"
uuid = "7e49a35a-f44a-4d26-94aa-eba1b4ca6b47"
version = "0.5.17"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.SciMLPublic]]
git-tree-sha1 = "0ba076dbdce87ba230fff48ca9bca62e1f345c9b"
uuid = "431bcebd-1456-4ced-9d72-93c2757fff0b"
version = "1.0.1"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.Setfield]]
deps = ["ConstructionBase", "Future", "MacroTools", "StaticArraysCore"]
git-tree-sha1 = "c5391c6ace3bc430ca630251d02ea9687169ca68"
uuid = "efcf1570-3423-57d1-acb7-fd33fddbac46"
version = "1.1.2"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
version = "1.10.0"

[[deps.SpecialFunctions]]
deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "f2685b435df2613e25fc10ad8c26dddb8640f547"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.6.1"

    [deps.SpecialFunctions.extensions]
    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"

    [deps.SpecialFunctions.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"

[[deps.StaticArrays]]
deps = ["LinearAlgebra", "PrecompileTools", "Random", "StaticArraysCore"]
git-tree-sha1 = "eee1b9ad8b29ef0d936e3ec9838c7ec089620308"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.9.16"

    [deps.StaticArrays.extensions]
    StaticArraysChainRulesCoreExt = "ChainRulesCore"
    StaticArraysStatisticsExt = "Statistics"

    [deps.StaticArrays.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[deps.StaticArraysCore]]
git-tree-sha1 = "6ab403037779dae8c514bad259f32a447262455a"
uuid = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
version = "1.4.4"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "7.2.1+1"

[[deps.SymbolicIndexingInterface]]
deps = ["Accessors", "ArrayInterface", "RuntimeGeneratedFunctions", "StaticArraysCore"]
git-tree-sha1 = "94c58884e013efff548002e8dc2fdd1cb74dfce5"
uuid = "2efcf032-c050-4f8e-a9bb-153293bab1f5"
version = "0.3.46"

    [deps.SymbolicIndexingInterface.extensions]
    SymbolicIndexingInterfacePrettyTablesExt = "PrettyTables"

    [deps.SymbolicIndexingInterface.weakdeps]
    PrettyTables = "08abe8d2-0d0c-5749-adfa-8a2ac140af0d"

[[deps.SymbolicLimits]]
deps = ["SymbolicUtils", "TermInterface"]
git-tree-sha1 = "5085671d2cba1eb02136a3d6661c583e801984c1"
uuid = "19f23fe9-fdab-4a78-91af-e7b7767979c3"
version = "1.1.0"

[[deps.SymbolicUtils]]
deps = ["AbstractTrees", "ArrayInterface", "Combinatorics", "ConstructionBase", "DataStructures", "DocStringExtensions", "DynamicPolynomials", "EnumX", "ExproniconLite", "LinearAlgebra", "MacroTools", "Moshi", "MultivariatePolynomials", "MutableArithmetics", "NaNMath", "PrecompileTools", "ReadOnlyArrays", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArraysCore", "SymbolicIndexingInterface", "TaskLocalValues", "TermInterface", "WeakCacheSets"]
git-tree-sha1 = "42e497496c1cac2280d523b39c95d41767996b4c"
uuid = "d1185830-fcd6-423d-90d6-eec64667417b"
version = "4.13.1"

    [deps.SymbolicUtils.extensions]
    SymbolicUtilsChainRulesCoreExt = "ChainRulesCore"
    SymbolicUtilsDistributionsExt = "Distributions"
    SymbolicUtilsLabelledArraysExt = "LabelledArrays"
    SymbolicUtilsReverseDiffExt = "ReverseDiff"

    [deps.SymbolicUtils.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
    LabelledArrays = "2ee39098-c373-598a-b85f-a56591580800"
    ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267"

[[deps.Symbolics]]
deps = ["ADTypes", "AbstractPlutoDingetjes", "ArrayInterface", "Bijections", "CommonWorldInvalidations", "ConstructionBase", "DataStructures", "DiffRules", "DocStringExtensions", "DomainSets", "DynamicPolynomials", "Libdl", "LinearAlgebra", "LogExpFunctions", "MacroTools", "Markdown", "Moshi", "MultivariatePolynomials", "MutableArithmetics", "NaNMath", "PrecompileTools", "Preferences", "Primes", "RecipesBase", "Reexport", "RuntimeGeneratedFunctions", "SciMLPublic", "Setfield", "SparseArrays", "SpecialFunctions", "StaticArraysCore", "SymbolicIndexingInterface", "SymbolicLimits", "SymbolicUtils", "TermInterface"]
git-tree-sha1 = "14818d4009715f19143964cc7994f8111e64f105"
uuid = "0c5d862f-8b57-4792-8d23-62f2024744c7"
version = "7.9.0"

    [deps.Symbolics.extensions]
    SymbolicsD3TreesExt = "D3Trees"
    SymbolicsDistributionsExt = "Distributions"
    SymbolicsForwardDiffExt = "ForwardDiff"
    SymbolicsGroebnerExt = "Groebner"
    SymbolicsLatexifyExt = ["Latexify", "LaTeXStrings"]
    SymbolicsLuxExt = "Lux"
    SymbolicsNemoExt = "Nemo"
    SymbolicsPreallocationToolsExt = ["PreallocationTools", "ForwardDiff"]
    SymbolicsSymPyExt = "SymPy"
    SymbolicsSymPyPythonCallExt = "SymPyPythonCall"

    [deps.Symbolics.weakdeps]
    D3Trees = "e3df1716-f71e-5df9-9e2d-98e193103c45"
    Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
    ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
    Groebner = "0b43b601-686d-58a3-8a1c-6623616c7cd4"
    LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
    Latexify = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
    Lux = "b2108857-7c20-44ae-9111-449ecde12c47"
    Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
    PreallocationTools = "d236fae5-4411-538c-8e31-a6e3d9e00b46"
    SymPy = "24249f21-da20-56a4-8eb1-6a02cf4ae2e6"
    SymPyPythonCall = "bc8888f7-b21e-4b7c-a06a-5d9c9496438c"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TaskLocalValues]]
git-tree-sha1 = "67e469338d9ce74fc578f7db1736a74d93a49eb8"
uuid = "ed4db957-447d-4319-bfb6-7fa9ae7ecf34"
version = "0.1.3"

[[deps.TermInterface]]
git-tree-sha1 = "d673e0aca9e46a2f63720201f55cc7b3e7169b16"
uuid = "8ea1fca8-c5ef-4a55-8b96-4e9afe9c9a3c"
version = "2.0.0"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.WeakCacheSets]]
git-tree-sha1 = "386050ae4353310d8ff9c228f83b1affca2f7f38"
uuid = "d30d5f5c-d141-4870-aa07-aabb0f5fe7d5"
version = "0.1.0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"
"""

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