FN3 Tweaks

[siwelwerd]

• Definition 2.3.2 refers to $y$-intercepts only in the singular
• Activity 2.3.3 Instead of "Graph A" and "Graph B", I wonder if this is a chance to reinforce the notation from the previous section (i.e. call one $f(x)$ and the other $g(x)$ or similar).
• I wonder if Activity 2.3.3(e) might be better as "Sketch a graph of a function with the following intercepts: " Then for part (f), repeat with two $y$-intercepts.
• Activity 2.3.11 Enlarge the open endpoint.
• In Definition 2.3.18 the statement "A relative maximum point is a point where the function changes direction from increasing to decreasing (making that point a "hill" in the graph)." is assuming continuity -- I think we should instead use a more general definition. And maybe also consider an activity with a function like $f(x)=|x|$ for nonzero x and $f(0)=1$.

[tdegeorge]

Are you saying that you want to write it like "intercept(s)" in Definition 2.3.2? I wrote both the x- and y-intercept as singular...

I still need help with Activity 2.3.11 - to enlarge the endpoint. I tried changing the size, but then it makes the endpoint not bold...I'm also having trouble with the next graph (Activity 2.3.13). The right endpoint should be open...

For Definition 2.3.18 - what would a more definition look like?

[tdegeorge]

Created a pull request because I did fix some of the mistakes, but will need to revisit this still.

[siwelwerd]

I think 2.3.11 looks good? I changed 2.3.13 to match it.

For 2.3.18, a maximum is a point where the function value is larger than all others in some neighborhood around the point. While I think the converse of your statement is true (if it switches from increasing to decreasing it's a max--that's the first derivative test), I am imagining some very wiggly function that has a minimum but all the wiggles mean we can't say it is increasing to the left (because it keeps wiggling).

[siwelwerd]

I marked #202 as draft since you still need to revisit this.

[siwelwerd]

Wiggles: I believe $f(x)=|x|+x\sin(1/x)$ (with $f(0)=0$) has a local minimum at zero, but there is no interval $(-a,0)$ on which $f$ is decreasing (and no interval $(0,a)$ on which $f(x)$ is increasing).