Graphing All the Way

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

A graph is just one of the three major representations (formula, table, graphing) of a relationship between variables. That is, it is one way to tell the tale of how one variable changes when the other increases. We are going to see how calculus can help us to pinpoint when certain “important events” occur on a graph and for the function in general.

For example, here is a graph of Harry’s position (in miles) from home (positive = east, negative = west) as he travels along an east-west road throughout the day (i.e., as time increases). Assume noon is when $t = 0$ hours. We can call Harry’s position function $f(t)$ . Although we have no formula for $f(t)$ , everything we will do here will correspond to algebraic methods we would do if we had a formula for his position (don’t worry- you’ll do these things with a formula soon enough!).

After each part, catch your breath and stop for a whole class discussion of the exciting things you’ve found.

For what times is Harry’s position east of home? West of home? What did you look for on the graph to answer those questions? What special thing happens at the demarcation times? If we had a formula for $f(t)$ , what would you do with the formula to determine those demarcation times? If you did not have the graph of $f(t)$ and only had the answers to the initial questions to this paragraph, what would you know and not know about the graph?

Now let’s examine Harry’s motion, not just where he is. For what times is Harry moving eastward? Moving westward? What did you look for on the graph to answer those questions? What special thing happens at the demarcation times? If we had a formula for $f(t)$ , what would you do with the formula to determine those demarcation times? If you did not have the graph of $f(t)$ and only had the answers to the initial questions to this and the previous exploration, what would you now know and not know about the graph?

Now let’s precisely examine Harry’s acceleration. When Harry is moving eastward, for what times is his speed increasing? For what times is his eastward speed decreasing? What did you look for on the graph to answer those questions? What special thing happens at the demarcation times? If we had a formula for $f(t)$ , what would you do with the formula to determine those demarcation times?

When Harry is moving westward, for what times is his speed increasing? For what times is his speed decreasing? What did you look for on the graph to answer those questions? What special thing happens at the demarcation times? If we had a formula for $f(t)$ , what would you do with the formula to determine those demarcation times?

Classify the four basic shapes of any part of a graph of a function by describing each shape in terms of what you found in the second, third, and fourth exploration..

Now do the following:

Sketch a graph of $f(x)$ with as much detail as possible given the following sign information about $f(x)$ , $f'(x)$ , and $f''(x)$ . Be sure to describe how you interpreted the information (e.g., what does knowing the sign of $f'$ on an interval tell you about the graph of $f$ on that interval?).

Up To x=	-3	1	4	5	7	9	10	Higher

f(x)	Neg	Pos	Pos	Pos	Neg	Neg	Pos	Pos

f’(x)	Pos	Pos	Neg	Neg	Neg	Pos	Pos	Pos

f”(x)	Neg	Neg	(Neg)	Pos	Pos	Pos	Pos	Neg

This just in: Other Fun Facts: $f(1) = 8$ , $f(4) = 2$ , $f(7) = -5$

For the graph of $f$ below, make a sign chart for $f(x)$ , $f'(x)$ , and $f''(x)$ (similar in form to that in $\#1$ ).

Determine whether the following properties can be satisfied by a function (of which you could draw the entire graph without lifting your pencil- which we call continuous). If such a function is possible, sketch a graph of that function. If such a function is not possible, explain why.

(a): A function $f$ is concave down and positive everywhere.
(b): A function $f$ is increasing and concave down everywhere.
(c): A function $f$ has exactly two extrema (i.e., mins and/or maxes) and three inflection points.
(d): A function $f$ has exactly four roots and two extrema (i.e., mins and/or maxes).

You are told that $f''(x) > 0$ for all $x$ . Which of the following must be true about the graph of $y = f(x)$ ?:

(a): The graph is a straight line.
(b): The graph crosses the $x$ -axis at most once.
(c): The graph is concave down.
(d): The graph crosses the $y$ -axis more than once.
(e): The graph is concave up.

You are told that $f''(x) > 0$ for all $x$ . Which of the following must be true about the maximum value of $f(x)$ on the domain $0 \leq x \leq 2$ ?

(a): Some critical value strictly between $0$ and $2$ .
(b): Either $x = 0$ or $x = 2$ .
(c): There is a maximum, but not enough information is given.
(d): $f$ need not have a maximum.

2024-10-10 13:50:45