From 9d2cbd8679c846531a0325036a2e66524dd70420 Mon Sep 17 00:00:00 2001 From: Kyle Spaans Date: Fri, 19 Jun 2009 10:58:22 -0400 Subject: [PATCH] almost done lec 20 notes --- lec20-0619.tex | 38 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 lec20-0619.tex diff --git a/lec20-0619.tex b/lec20-0619.tex new file mode 100644 index 0000000..2b361b9 --- /dev/null +++ b/lec20-0619.tex @@ -0,0 +1,38 @@ +\documentclass{article} +\usepackage{fullpage} +\usepackage{amsmath} +\author{Kyle Spaans} +\date{June 19, 2009} +\title{Calculus 3 Lecture Notes} +\begin{document} +\maketitle + +\section*{Lecture 20 -- Global Extrema and Optimizing With Contraints} +We finish the example from Wednesday, emphasising the importance of memorizing +trigonometric angles and values, which will be very important when we get to +triple integrals. + +\paragraph*{Example 1} +Parameterize $x = \cos t$, $y = \sin t$. Compute +$\frac{d}{dt} f(x(t), y(t)) = 30\cos t (\cos^2 t -3\sin^2 t) = 0$ +And solving that to find where it is equal to 0 will give us the extrema. We +get values for $t$ of $\frac{\pi}{6}, \frac{5\pi}{6}, ...$ This give us a +value to plug into $f$ to findd the max: $f(\frac{\sqrt{3}}{2}, \frac{-1}{2})$ + +\paragraph*{Example 2} +Find the maximum and minimum values of +$f(x,y) 4xy - x^2 - y^2 - 6x$ +on the domain $S$ where $0 \le x \le 2, 0 \le y \le 3x$. Drawing this domain, +we get a triangle in quadrant 1, right angle at (2,0) and with a sloped side +made from the line $y = 3x$. $S$ is closed and bounded, $f(x,y)$ is continuous +by inspection, therefore the max and min exist. Now we can plug in our +algorithm. +$\nabla f = (f_x, f_y) = (4y - 2x - 6, 4x - 2y) = \vec{0}$ +And we can easily substitute $y = 2x$ into $f_x$ to find $x=1, y=2$ as a +critical point. Next, we examine each of the boundaries, considering them as +lines. Sub in $(x,0)$ ... + +\subsection*{Optimization with Contraints} +Find the maximum and/or minimum of $f(\vec{x})$ subject to contraint +$g(\vec{x}) = k$ (with $k$ constant)... +\end{document} -- 2.11.0