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.
- Know the different notations 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.
- 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.
- 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 what the
theorem says and you need to know how to apply it to find
solutions of equations in a given interval.
- 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 function is continuous and 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 me 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, steady states or whatever else. If you know the above
this 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 section number 48, 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.
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, espcially in the
graph. If you want to mark an inflection point, make 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.