Please note that my Midterm Exam Prep Seminar for Math 1500 is coming soon (Part 1 is Sunday, Oct. 13 and Part 2 is Sunday, Oct. 20).  Even if you are distance, you will find this seminar quite a helpful review of the first half of the course.  Please note that I will not discuss the topics covered in this seminar at the Final Exam Seminar in December.  For more info about the seminar, and to sign up if you wish, please click this link:

Did you read my tips on how to study and learn Math 1500?  If not, here is a link to those important suggestions:

Did you see my tips for Assignment 1?  If not, here is a link to those important suggestions:

These are tips for the second assignment in the Distance/Online Math 1500 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.

Here is a link to the actual assignment, in case you don't have it:

Note that you need to study Lesson 2 (Limits), Lesson 3 (Continuity), Lesson 4 (The Definition of Derivative), and Lesson 5 (The Differentiation Rules) from my Intro Calculus book to prepare for this assignment.

Don't have my book? You can download a sample containing lessons1 and 2 here:

This question uses definitions I don't discuss in my book but I think it is kind of obvious which graph is which here (there is one of each).  If you look at the graph I drew at the start of my Limits Lesson on page 32, x = -1 is a removable discontinuity because we could "fix" that discontinuity by filling that hole with a dot; x = 2 is a jump discontinuity because the graph jumps from one location (y = 4) to an entirely new location (y = -1); x = 4 is an infinite discontinuity because the graph is flying away to infinity or negative infinity as it approaches 4.

This question should be solved by using the Three Steps to Check Continuity I teach in Lesson 3 to prove there is a discontinuity at x = 2, as they suggest.  Then, by visualizing what the limits tell you about the graph, you should be able to identify the type of discontinuity you are dealing with.

Note that those strange double square brackets around the function in part (a) denote the greatest integer function.

[[x]] is computing the greatest integer of x.  Visualize a number line where you have marked off all the integers: ... -3, -2, -1, 0, 1, 2, 3 ...  Place x on that number line, [[x]] is the nearest integer to x that is no larger than x itself (the integer immediately to the left of x).

For example:
        [[ 2.6 ]] = 2 since 2.6 lies between 2 and 3 on the number line
        [[ -5.2 ]] = -6 since -5.2 lies between -6 and -5 on the number line.
        [[ 4 ]] = 4 since 4 lies exactly on 4 on the number line.

Note that you are not rounding off the numbers when you compute the greatest integer, you are always stating the integer that is immediately to the left of the given number (the nearest integer that is less than or equal to the given number).

The graphs of greatest integer functions tend to be "step" graphs.  A series of horizontal lines jumping from one horizontal step to another.

For example, [[ x ]] would have a horizontal line at y = 0 for the region [0, 1), with a dot at (0,0) and a hole at (1,0); then jump to a horizontal line at y = 1 for the region [1, 2) with a dot at (1,1) and a hole at (2,1); then jump to a horizontal line at y = 2 for [2, 3) with a dot at (2,2) and a hole at (3,2); and so on.

This question is using the Intermediate Value Theorem as illustrated in questions 4 and 5 in Lesson 3.

This is a classic continuity problem as discussed in questions 1, 2 and 3 of my Lesson 3.

This is infinite limits as discussed in questions 12 to 14 in Lesson 2.  Make sure you look at my Practise Problems in that lesson, too.  Especially question 74.

This is similar to my question 15 in Lesson 2.  You must find all the values of x that cause K/0 to find the Vertical Asymptotes, and compute the limits as x approaches both positive infinity and negative infinity to find the Horizontal Asymptotes.  Note that there can be more than one of either asymptote.

Technically, you can just use the differentiation rules to find the derivative in this question, but, judging from the sections in the text they are referring you to, I think they want you to use the Definition of Derivative to find the derivative in this problem.  Study Lesson 4 in my book to learn how to use the definition of derivative and find the equation of a tangent line.

Another Definition of Derivative problem as taught in Lesson 4.

A good run-through of the various Differentiation Rules, as taught in Lesson 5 of my book.

I show you how to solve trig limits such as this back in Lesson 2, question 16 of my book.