# Topics on Test #2

1) LHopital’s Rule

2) Related Rates

3) Implicit Differentiation

4) Optimizations

5) Anti-derivatives

6) Curve Sketching

7) Extreme Values

8) Monotonicity

9) Shape Of Graphs

These are the topics I can think of on the top of my head.

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### 5 Responses to Topics on Test #2

1. Kate Poirier says:

Looks pretty good to me! Keep in mind that I organized some things a little differently from how your text and WebWork did. For example, your items 6, 7, 8, and 9 all fall under the heading “shapes of graphs” for me. This includes the list of all possible features of a graph that can be obtained by analyzing the formula for the function itself, as well as its first and second derivatives.

A good exercise would be to generate a sub-list of topics that fall under this broader “shape of graphs” heading. Either as a comment on this post, or as a new post altogether, I’d like one of you (Jimmy and/or anyone else) to provide a list of every feature of the graph of a function (that you’ve learned about in this course) that you can use to give a “reasonably accurate” sketch of the graph. Hint: I gave you a list in class way back when we started talking about these topics.

I will definitely be asking you at least one question on your next test where you have to sketch a graph by hand using each of the items on this sub-list. Whether I ask you for the individual features or not, this sub-list is your to-do list when it comes to sketching graphs of functions. So you’ll need to check off all of the items, even if the instructions for a question are simply, “sketch the graph of the function.”

2. Maloney says:

A list for sketching:
Domain, x intercepts, y intercept, end behavior (horizontal asymptotes etc.)
Discontinuities: vertical asymptotes, removable discontinuities, etc? + nearby behavior.
f(x) Sign analysis: Sign of the function over intervals.
f ‘ (x) Sign analysis: critical points, Local Maxima/Minima.
f ” (x) Sign analysis: Inflection points and concavity.
Bonus: Consider symmetry to save work: even f(-x) = f(x); odd f(-x) = -f(x).

3. Kate Poirier says:

This looks good! The classification of discontinuities and the behavior of a function near a discontinuity are essentially two sides of the same coin. Remember that studying limits as x approaches a discontinuity from either side will tell you what kind of discontinuity you’re dealing with.

More specifically…

If the limit from either side is $\pm \infty$, then the graph has a vertical asymptote; the $+$ or $-$ sign will tell you whether the function shoots up or down.

If the limits from either side are both finite, but not equal, then you have a jump discontinuity.

If the limits from either side are finite and equal, but do not equal the value of the function at the point, then you have a removable discontinuity (hole in the graph).

If either the left-hand or right-hand limit fails to exist (neither finite nor infinite), then there’s a different kind of discontinuity. An example is the function $f(x)=sin(\frac{1}{x})$ near $x=0$.

4. Ismail Akram says:

So if study this list, I’m all good for the test?

5. Kate Poirier says:

Hi Izzy, I just saw this now. My answer would have been yes! After taking the test, I hope that’s your answer too.