Reading Questions that were asked.
This week I’m trying something different on the reading questions. I am going to
group the questions by “questions that will help you as you read” and then by
“questions that are more directed at me.” One of my goals (as always) will be for
you to start asking more questions of the first type, but I will respond to all of them
here.
Questions that will help you as you read.
Does Fermat's Theorem refer to the fact that to find the max or mins of a function, you
have to take the derivative and find the zeroes?
Fermat’s Theorem is the statement that if the derivative of a function exists at a
point that is a local extremum, then the derivative at that point must be 0. This
means that a method for finding the local maxima and minima is to
1. Find the x-values where the derivative is equal to 0,
2. Find the x-values where the derivative doesn’t exist,
3. Evaluate the function at all of the x-values in the first two steps as well as at
the end points.
4. Classify each of these values as a local max, local min, global max, or global
min.
If a function is convex down, wouldn't it always be decreasing? (same with convex up
and increasing)
This is one of the things that makes concavity confusing. A function can be
concave down and increasing as in the following graph.
3.0
2.5
2.0
1.5
1.0
0.5
2
4
6
8
10
In this graph, the graph is concave down, but the function is increasing. The
derivative of the function, however, is decreasing.
Similarly, if the graph is concave up, it could still be decreasing (like the
graph of y=1/x (for x>0).
How do you find the endpoints of the domain of a function when they aren't given?
In a standard textbook problem, if they aren’t given, the domain is assumed to be
the whole real line (or where the function is defined). In the type of problem that
you will encounter in the life sciences, the endpoints are usually determined by
context. If the independent variable is time (for example) you would want the
endpoints to be the times that you care about for your application.
If there is a hole in a graph, is there an asymptote through it? Or is it just an isolated
incident?
I’m not entirely sure about what is meant in this question. If the right hand and
left had limits at the x-value of the hole are infinity, then there is an asymptote. If
the limits are finite, then it is just an isolated point of discontinuity.
Seeing that the second derivative tests concavity and also helps in finding the inflection
point by comparing the value to 0, would it also be possible to use the technique of
substituting either a positive or negative value to determine concavity in the same way
it is possible to apply the same technique to determine whether a function is increasing
or decreasing at a given point?
Also, are critical numbers identical across all
derivative tests?
Yes, this is indeed a way to determine convexity. That is, once you have
identified where the second derivative is 0 or does not exist (be careful that you find
both), then substituting in values in each of the intervals can tell you what the
convexity is inside of that interval. Similarly, you can do this for increasing and
decreasing with the first derivative.
I’m not sure I understand the last question. The critical points are precisely
those values where the first derivative is either 0 or does not exist.
If you are given a function with a set domain, does that mean you ignore all values
outside of the given x-value range?
In terms of a textbook question, the answer is yes. If it is a function that comes
up in a life sciences application, then you would ignore values outside the range that
you are interested in. However, if there are interesting values outside that range,
you might see if that means that you should be interested in more than you
originally thought.
Can positive or negative infinity be a local extrema?
We don’t call positive and negative infinity either local extrema. If the function
has a limit of positive infinity, we say that it has no maximum value or is unbounded
in the positive direction and if it has a limit of negative infinity, we say it has no
minimum value and is unbounded in the negative direction.
Does the third derivative of x indicate anything useful about the graph of the
function?
Yes, although we usually don’t discuss it in MATH 122. The third derivative
tells you fast the second derivative is changing (sometimes called the “jerk”). A
large positive third derivative means that the second derivative is increasing
quickly.
How do you find local extrema?
By Fermat’s theorem, you find them by looking at where the first derivative
is either 0 or does not exist. Once you know that, you can see whether the graph is
increasing on one side and decreasing on the other (usually by using the first
derivative). In general, if the first derivative is
Neg 0 neg
\
\__
neg 0 pos
\
pos 0 neg
/
\__/
\
\
neither
pos 0 pos
___
/
/
loc. Min
/
__/
\
\
local max
/
/
neither
Is concavity only used to determine acceleration or are there other applications?
It has lots of applications. Concavity is used to tell you how the rate of
change is changing. Thus, if we are looking at the population of the US, the
population is continuing to increase, but the growth of the population is slowing.
This slowing is shown in the concavity (2nd derivative is negative). Anytime you are
thinking about the change of a growth rate, you are thinking about the second
derivative.
Where does a function change from being concave down to concave up or vice versa if
it is continuous and goes from one extrema to the other?
A function changes concavity at an inflection point (the definition of an
inflection point). Inflection points occur where the second derivative is 0 or does
not exist. Unfortunately, there is no simple way to find this without using the 2nd
derivative).
Are the extreme values in the interior of a function just the function's endpoints? They can occur at the functions endpoints, but the extreme values are the
maximum and minimum value of the function. Thus in the example below, the
maximum value is 16 and occurs at x=-2, while the minimum value is -16 and occurs
at x=2.
Why is the derivative of the local extremum at an internal point always 0? (fermat's
theorem)
The derivative is 0 only if the derivative exists. Let us start by just looking at
a local max. If the function y=f(x) has a local maximum at x=a, that means that the
function must be getting bigger when x is just a little bit less than a, and it must be
getting smaller when x is just a little bit more than a. But the function is increasing if
the derivative is positive (and exists) and is decreasing if the derivative is negative
(and exists). Thus the derivative must go from positive to negative when it crosses
x=a. Consequently, at the point x=a, the derivative must either be 0 (so it can cross
from positive to negative) or not exists (so it could potentially jump over). There is
a very slight lie above, in that the function could have derivative 0 at a point and still
be decreasing or increasing as in the ( neg 0 neg ) example or ( pos 0 pos ) example
four questions above.
How do people determine which part of a function is the most important to determine
an interval for?
This is usually based on context. If you are worried about bacteria growth in
the first 24 hours after a treatment, then you are probably only worried about the
interval t=0 to t=24, although you might be worried about an interval starting
slightly earlier if it turns out that you want to know how much the treatment has
slowed the growth… If I am examining a function without a context, then I typically
am worried about where the “interesting” things are happening in the function,
which means I will often look for the local max, local min, inflection points, etc…
The book doesn't state the defintion of convex up and down. Are they the opposite or
the same as concave up and down?
Hmm…. I think I owe you all a donut point. I should have said concave up
and concave down not convex up and convex down. We do use both, but concave is
the right term… 
Functions like that in Question 1 where x is raised to the third power, do these
functions never have global extrema only local extrema since they are increasing or
decreasing infinitely?
If the domain is the entire real line then any polynomial with an odd degree
will never have global extrema for this reason.
When a functin is increasing/decreasing, are you asking with respect to x or y axis? for
ex, #1, isnt the entire function increasing?
In example #1, the function is increasing throughout the entire domain,
(which is the mathematically correct way to say what you are getting at). I’m not
sure what you mean in terms of the x-axis or y-axis though. We think of x as the
independent variable in this case.
Are domains a necessary component to figuring out the graph or am I wrong to say I
cant find those things because there is another way without requiring an x or a
domain?
I’m not sure what you mean. In mathematics, the domain is given to you with
a function. If you aren’t given an explicit domain, then we assume the domain is the
set of all x-values that can be used for the function. Thus in Ex. 1 of the reading, the
domain would be the whole real line if it is not given to you.
Questions directed towards me. (I will put in parentheses what I would
consider a more helpful version of the question as well as endeavor to answer
it.)
I’m still a little confused on the local vs. global extrema. Could you explain this one
more time in class please?
(More helpful question for reading comprehension: What is the difference
between local and global extrema?)
Local extrema are the places where the graph of the function has a low point
(water would collect in) or a high point (water falls off to either side). Global
extrema are the highest of the high points and the lowest of the low points. In the
graph below, there are local minima at x-values close to -1.5 and .7, with associated
y-values of about -3.7 and -1.4. There is a local maximum at x=0 with associated yvalue of -1. The Global minimum is -3.7 (at approximately x=-1.5). The global
minimum for the given domain is at x=1.2 and is about y=0.
I am really not understanding the difference between monotonicty and
concavity? What is the difference? Is it based on a 'strict inequality'?
(More helpful question for reading comprehension:What is the difference
between monotonicity and concavity? Is the difference based on strict
inequality? If not, what is the strict inequality tell us?)
Monotonicity just tells us that a function is either increasing on the entire
domain or decreasing on the entire domain. It has nothing to do with concavity.
The strictness issue is about whether the function is always increasing (strict) or
sometimes it is flat on a piece of the domain (not strict) and similarly for decreasing.
Thus a function is monotone increasing if the derivative is always positive or 0, and
it is monotone decreasing if the derivative is always negative or 0. Concavity tells us
whether the derivative of the function is increasing or decreasing. Thus, the
function y=x^2 is not monotone, since the function decreases for negative x values
and increases for positive x values. However, the function is concave up since the
derivative (dy/dx= 2x) is always increasing.
Questions not on the reading.
Why isn't the inverse of x^2 the square root of x? I did not understand this too clearly
in class
If we restrict the domain of the function y=x^2 to positive x-values, then the
inverse would indeed by the square root of x. However, the square root only gives
positive x values, and by convention and notation, we have
√(−2)2 = √4 = 2. However, if it was an inverse function, it would need to equal -2.
I don't know if you still want us to ask a question about the project but how will the
paper be graded? Will it be with the same rubric as the video?
My thought was that I would use the rubric on both the video and paper
portion. My hope is that the paper portion will provide references and details
where the video might not have them.
Unrelated to the reading, will our next test be pushed back from the date on the
syllabus like our last one was?
No, as announced in class last week, the test will be next Tuesday, 11/19.
As for the video project I think other students should be able to watch one other video.
I will send out a survey on this to get the full class input, but thank you for
the reminder.