Stuck in the Middle

$\newenvironment {prompt}{}{} \newcommand {\pKa }{\text {p}K_{\text {z}}} \newcommand {\pKb }{\text {p}K_{\text {b}}} \newcommand {\Ka }{K_{\text {z}}} \newcommand {\Kb }{K_{\text {b}}} \newcommand {\Kc }{K_{\text {e}}} \newcommand {\Ke }{K_{\text {e}}} \newcommand {\Kw }{K_{\text {w}}} \newcommand {\pH }{\text {pH}} \newcommand {\pOH }{\text {pOH}} \newcommand {\ungraded }[0]{} \newcommand {\framed }[0]{\MakeFramed {\advance \hsize -\width \FrameRestore }} \newcommand {\shaded }[0]{\def \FrameCommand {\fboxsep =\FrameSep \colorbox {shadecolor}}\MakeFramed {\FrameRestore }} \newcommand {\leftbar }[0]{\def \FrameCommand {\vrule width 3pt \hspace {10pt}}\MakeFramed {\advance \hsize -\width \FrameRestore }} \newcommand {\titled-frame }[1]{\def \FrameCommand {\fboxsep 8pt\fboxrule 2pt \TitleBarFrame {\textbf {#1}}}\def \FirstFrameCommand {\fboxsep 8pt\fboxrule 2pt \TitleBarFrame [$\blacktriangleright $]{\textbf {#1}}}\def \MidFrameCommand {\fboxsep 8pt\fboxrule 2pt \TitleBarFrame [$\blacktriangleright $]{\textbf {#1\ (cont)}}}\def \LastFrameCommand {\fboxsep 8pt\fboxrule 2pt \TitleBarFrame {\textbf {#1\ (cont)}}}\MakeFramed {\advance \hsize -20pt \FrameRestore }} \newcommand {\FrameCommand }[0]{\setlength \fboxrule {\FrameRule }\setlength \fboxsep {\FrameSep }\fbox } \newcommand {\FirstFrameCommand }[0]{\FrameCommand } \newcommand {\MidFrameCommand }[0]{\FrameCommand } \newcommand {\LastFrameCommand }[0]{\FrameCommand } \newcommand {\FrameHeightAdjust }[0]{6pt} \newcommand {\instructorNotes }[0]{\setbox 0\vbox \bgroup } \newcommand {\instructorIntro }[0]{\setbox 0\vbox \bgroup } \newcommand {\teachingnotes }[0]{}$

We now know how to find the exact (or at least an estimate) of the area between a graph of a function and the $x$ -axis. For most of our problems, this has been interpreted as the net change between two times for a given rate function, although it could be treated as a purely geometrical measurement as well.

Here, we expand what we know to finding the area between two curves (with neither of them being the $x$ -axis).

See if you can come up with a technique for finding the area of the region bounded by the graphs of $f(x) = 2x^2-6x+13$ and $g(x) = x^2-2x+34$ . Then find the exact area. (You might want to graph the region first to give you an idea of what to do).

What is different about the problem if we are finding the area of the region bounded by $f(x) = x^2-7x+19$ and $g(x) = x^2-4x-13$ on $[-1, 5]$ ? How does your technique change? Find the exact area.

What is different about the problem if we are finding the area between $f(x) = 2x^2-6x+13$ and $g(x) = x^2-2x+34$ on $[-3, 10]$ ? How does your technique change? What two math questions could we ask here?

Go back to the race between the chicken and the goat in Homework #6. Graph the velocity functions on the same axes. What does finding the net area between the graphs tell us about the two animals and the race?

2024-10-10 13:53:23