7.5: Summary
- Page ID
- 121204
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Optimization is a process of finding critical points, and identifying local and global maxima/minima.
- A scientific problem that address "biggest/smallest, best, most efficient" is often reducible to an optimization problem.
- As with all mathematical models, translating scientific observations and reasonable assumptions into mathematical terms is an important first step.
- The following applications were considered:
- Density dependent population growth. Using a given logistic growth law, the following parameters were considered:
- population growth rate (to be maximized),
- population density,
- intrinsic growth rate (constant),
- carrying capacity (constant).
- Nutrient absorption in a cell. Using the model developed in Section \(1.2\) for a spherical cell, we considered:
- nutrient absorption rate,
- nutrient consumption rate,
- cell radius,
- proportionality constants (determined based on context). We maximized the net rate of increase of nutrients - a difference between absorption and consumption rates.
- Surface area of a cylindrical cell, which tends to be minimized do to energy conditions. The parameters we used were:
- cell length,
- cell radius,
- cell volume (constant),
- cell surface area (to be minimized).
- Wine for Kepler’s wedding, seeking the largest barrel volume for a fixed diagonal length. The following parameters were considered:
- barrel volume, (to be maximized)
- barrel height,
- barrel radius,
- length of the diagonal (constant).
- Foraging time for an animal collecting food. We considered:
- travel time between nest and food patch,
- foraging time in the patch,
- energy gained by foraging in a patch for various time durations.
- Density dependent population growth. Using a given logistic growth law, the following parameters were considered:
- If the growth rate of a population follows the following logistic equation:
\[G(N)=1.2 N\left(\frac{50000-N}{50000}\right), \nonumber \]
where \(N\) is the density of the population, under what circumstances is the population growing fastest?
- When finding a global maximum, why is always imperative to check the endpoints?
- Demonstrate the variability of barrel dimensions by giving two different height and radius pairs which lead to a volume of \(50 \mathrm{~L}\).
- Would the answer to Kepler’s wine barrel problem have changed if we had solved for \(h^{2}\) instead of \(r^{2}\) ?