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My Midterm Exam Prep Seminar for Math 1500 is Sunday, February 22. It is in room 100, St. Paul's College, 9:00 to 6:00 pm. It costs $40 for everyone (whether or not you bring my book). You pay at the door cash or cheque.
Even
those of you taking the course by distance (and who therefore do not have a midterm exam) should consider attending the seminar since it will be a good review of the course so-far. It will also be the only time I will discuss Lessons 1 to 7 in my book (there is no time to discuss these again at the final exam seminar).
Did you read my tips on how to study and
learn Math 1500? 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. The first assignment is a great way to build and review key skills that will be helpful for this course. Here is a link to the actual assignment, in case you don't have it: 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 or audio? You can download a free sample of my book and audio lectures containing Lessons 1 and 2: 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. There is certainly a typo in this question. There is absolutely nothing wrong with the function given in part (b), and it is continuous at -1. I believe that it is a typo and that they meant that part to be the greatest integer
function. CONTACT THE PROF and notify him that something is wrong with part (b).
In general, Question 2 can be solved if you use the three steps to check continuity and then visualize what the limits tell you about the graph.
- If the limit exists (which is to say, if the limit is the same from both sides), yet the function is discontinuous at x = -1 (perhaps because f(-1) is undefined), that
would indicate the graph connects to a hole from both sides. That is a removable discontinuity.
- If the limit goes to infinity or negative infinity on either side, that is obviously an infinite discontinuity.
- If the limit goes to one finite value as you approach from the left, but a different finite value as you approach from the right, that is a jump discontinuity.
I believe they
meant to put double square brackets around the function in part (b) to denote the greatest integer function. Again, contact the prof to find out what part (b) should really say.
I assume they meant to write in part (b): [[x+1]] which means the greatest integer of x+1.
For example:
[[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 closest integer that is less than or equal to x).
Here are some examples:
- [[ 2.6 ]] = 2 since 2.6 lies between 2 and 3 on the number line. 2 is the nearest
integer that is less than or equal to 2.6.
- [[ -5.2 ]] = -6 since -5.2 lies between -6 and -5 on the number line. -6 is the nearest integer that is less than or equal to -5.2.
- [[ 4 ]] = 4 since 4 lies exactly on 4 on the number line. 4 is the nearest integer that is less than or equal to 4.
The graphs of greatest integer functions tend to be "step" graphs. A series of horizontal lines jumping from one
horizontal step to another.
[[ x ]] would have a horizontal line at y = 0 for the region [0, 1), then jump to a horizontal line at y = 1 for the region [1, 2), then jump to a horizontal line at y = 2 for [2, 3), and so on.
This question is using the Intermediate Value Theorem as illustrated in Lesson 3, questions 4 and 5. This is a classic continuity problem, as illustrated in my Lesson 3, questions 1, 2 and, especially, 3.
This is infinite limits as discussed in Lesson 2, questions 12 to 14. Make sure you look at my Practise Problems in that lesson, too. Especially question 74.
Make sure that, when you are computing the square
root of x-squared that you include a note on the side explaining that the square root of x-squared is the absolute value of x. Then determine if that means the absolute value of x is x or -x, as the sign of x dictates in the particular limit. You must include this note in your answer!
This is similar to my Lesson 2, question 15. You must find all the values of x that cause K/0 to find the Vertical Asymptotes (i.e., look for bottom zeros), 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.
Athough they don't make it clear, the sections they are referring to in the text suggest 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.
You can check that your answer for the derivative is correct by using the differentiation rules. You will have to simplify the derivative in order to confirm that it matches your previous answer. Another Definition of Derivative problem as taught in Lesson 4.
You can check that your answer for the derivative is correct by using the differentiation rules. You will have to simplify the derivative in order to confirm that it matches your previous
answer. 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.
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