Final Exam Review

The Final Exam Review questions are here:

Final Exam Review.


Please select some that you would like to put on the board on Monday. Tomorrow I will ask you to sign up for the problem of your choice.

But prepare as many as possible, in case someone has not signed up for a problem or the person is absent.

Sketching a complete graph using what we know from Calculus


A complete graph of a function must include all of the following if they exist:

  • The x- and y-intercepts
  • Local maxima and minima (turning points)
  • Concavity and points of inflection
  • Vertical asymptotes
  • Horizontal asymptotes
  • The end behavior of the graph (behavior as x goes to infinity  or – infinity) – horizontal asymptotes, if they exist, are one form of end behavior.
  • The domain of the function should be clear from the graph

Exception: in the case of periodic functions, we often sketch only one period of the function.


Comments on some of  these:

The y-intercept is the point where x=0, if 0 is in the domain of the function.

The x-intercepts are the zeroes of the function: set y=0 and solve.

There is a vertical asymptote x=c when c is not in the domain of the function, and at least one of the following is true:

  • \displaystyle \lim_{x\rightarrow c}f(x) = \infty or -\infty
  • \displaystyle \lim_{x\rightarrow c^{+}f(x) = \inftyor -\infty
  • \displaystyle \lim_{x\rightarrow c^{-}f(x) = \inftyor -\infty

There is a horizontal asymptote y=c if

\displaystyle \lim_{x\rightarrow\infty}f(x) = c

The end behavior of a polynomial is determined by its leading term. See Session 9 in the Precalculus  textbook by Carley and Tradler.



Test 3 review UPDATED

Test 3 is scheduled for the first hour or so of class on Wednesday 25 April.


Review problems are here:


Answers and some hints are here: UPDATE The answer to question #3 has been corrected!


Please let us know on Piazza if you find any typographical or other errors in these!


This test will include the following topics:

• Derivatives of inverse trig functions

• L’Hospital’s Rule

• Absolute maxima and minima on a closed interval

• Intervals of increase and decrease, and local maxima and minima

• Intervals of concavity and inflection points



Important notice about midterm grades!

Midterm grades were sent to your citytech email address. Make sure that you look for them as the deadline to withdraw if you choose to do so is tomorrow!

For three students, the email was returned to me as undeliverable. I have been unable to solve the problem. These students are:

Hadiyah James

Asim Razzaq

Amal Ulla

I have posted your midterm grades to you as a private note in Piazza, if you activated your Piazza account. One person listed above does not have an active Piazza account, so I am unable to send the grades to you in any way whatever. Please contact me as soon as possible using an email address other than your citytech email, so that I can send you your midterm grades!

Homework for Wednesday 18 April

• Review the methods for finding absolute maximum or minimum in a closed interval, and the First and Second Derivative tests. My notes are linked in Monday’s post

• Finish the WeBWorK “Extreme Values”. Make sure that you use the First Derivative Test when the problem asks for local maxima or local minima.

NOTE: There appears to be a bug in the scoring of problem #9, where it marks answers for the local max or local min incorrect if you include the absolute max or absolute min. THE SCORING IN WEBWORK IS WRONG. Please just enter the answer that you think is correct, ignore the scoring on the local max and local min parts, and I will score these by hand when the assignment closes.

• Do the WeBWorK “Extreme Values – First and Second Derivative Tests” making sure that you use the method that is called for in each problem!

There will be a Quiz on Wednesday based on one of the problems in the WeBWorK “Extreme Values”. It will ask you to find local maxima and local minima as well, so make sure that you understand how to use the First Derivative Test to do this!

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

Monday 16 April class


• Review of the method for finding absolute maximum or minimum values on a closed interval. (Problem 4 from the WeBWorK “ExtremeValues”)

• The First Derivative Test for a local maximum or  minimum

• Concavity of the graph and the second derivative

• The Second Derivative Test for a local maximum or minimum

It is a very good idea to try out using both the First Derivative Test and the Second Derivative Test on the same problem, when you are starting out. The Second Derivative Test can be quicker, but it fails when the second derivative is 0. See which one you prefer. (But make sure you know who to use both of them!)

Here are my notes:


(Contains notes on the closed interval method as well as the First and Second Derivative tests.)


(Shows examples of concave up and concave down)