Study Guide for the 2nd midterm
Differentiation Rules
- You need to know the rules of differentiation and apply them
correctly and efficiently. Expect to have a number of differentations to
apply the many rules we have learned during the semester. No proofs will be
included in the exam.
- Learn the different notation for derivatives (dy/dx, y', f'(x) and so on). I could
use any of them.
- Higher derivatives and iteration of differentiation. We could ask you to calculate
the nth derivative of a function. Learn the notation also.
Derivatives of trigonometric functions and chain rule
- Learn how to differentiate trigonometric functions
and inverse trigonometric functions.
No proofs included.
- Chain rule. Very important, of course, you will probably use it more
than once in the exam. By now you should be very comfortable with this rule. If
you are not, try to practice as much as you can.
- Related rates problems are also included here: if a function depends
on another one and the second one has a certain rate of change, what is the rate of
change of the first one. Some problems are in this section, and one explanatory box.
- Implicit differentiation as an application of the chain rule. You could
be asked explicitly to use implicit differentiation even if you can solve for the function,
so don't trust you will be able to solve for it.
Graph sketching
This is a large chapter with many different topics. It is also a good one to usefor the TRUE/FALSE question, although I could use any of the sections.
- You need to know how to find the equation of the tangent and normal lines to
the graph of a function at a given points.
- The intermediate Value Theorem: you need to know the statement of the theorem (precisely,
we will take points if simply sketched, see the last comment) and you need to know
how to apply it to find solutions of equations in a given interval.
We use this theorem to study when a function is above or below the axis, but not much more here.
- Know how to figure out when a function is increasing or decreasing using the derivative.
- The Mean value Theorem: you need to learn its statement, it is a very important
theorem and you should know what it says.
- Stationary points: what they are and the different types.
- Maxima and minima: local max and min and how to find them using the derivative (the first and the second derivative). Global max
and min and how to find them: to find the global ones you need to find local ones and the values
of the function at the border of the domain. If the domain is infinite, you need to calculate
the limit. Once you have all these values, you pick the largest or the smallest. Know that,
if the domain is a closed interval, the function will always have a global max or min, while,
if the domain is not a closed interval they might not exist. The value needs to be reached
for some x to be a max or min.
- Finally: convexity, concavity, infection points and how to use the second derivative
to study all these. You might get this part as a portion of a graph, or by itself.
- Sketch the graph of a function: you will certainly get one in the exam. This question
might have specific questions about max, min, increasing, decreasing, convex, concave, inflection
points, stationary states or whatever else. If you know all of the above this portion should not be a problem.
Remember to check the domain and draw vertical lines that the graph cannot touch, if they
exist. Remember also to take limits at infinity or at points that are not in the domain.
Optimization problem
Not much here other than solving problems where one has to find the max or min
of a certain quatity under certain conditions. Students often have difficulties
with these problems because they are word problems that one needs to understand and
think about. There is only one way to learn these: practice, practice and then practice
some more.
Exponential and logarithms
- You need to know the properties of logs and exp well enough to be comfortable
using them. If you are not comfortable, please practice. This is pre-Calculus material
and we will penalize not knowing it.
- Find derivatives of exponential and logarithms, perhaps in combination with
chain rules and other functions.
- Learn the limit properties of log and exp when compared to powers of x. This
is Chapter 6, section 7, we expect you to know how to use these limits in perhaps more
complicated limits. Look at the homework to have an idea of what I mean.
- Exponential growth and decay: not much here, we might give you a problem to find
X_0 or k, or you might be given these values and asked something else. Again, homework
tells you what is going on here.
Notation
As in the first midterm, we ask you to write math properly. Do not skip equal signs,
fractions, etc. We can't read your mind and we will not make an effort to do so.
If it is not on paper, we assume it is not there and you did not know. The more you tell us about what you know, the more you might get extra points (if correct, of course).
We will have higher expectations this time around: you have had time to
practice and we expect you will be better at writing. Last exam there were mistakes
in writing that we did not penalize, but we will penalize it now. So be careful and
try your best, especially in the graph. If you want to mark an inflection point, me
sure to mark it. Having a curve that is vaguely pointing up is not enough to convince
us that you knew the graph was convex there. Write convex with an arrow pointing at
it, then we are sure you know. Same with the other properties, I hope you get the point.