Even if you are not taking the distance course, I think it is very useful for all Math 1500 students to attempt these hand-in assignments. In general, the assignments can be quite demanding and really force you to solidify your math skills.
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 the first two
lessons here:
Question 1 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.
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. 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.
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.
Question 3 is using the Intermediate Value Theorem as illustrated in questions 4 and 5 in Lesson 3.
Question 4 is a classic continuity problem as discussed in questions 1, 2 and 3 of my Lesson 3.
Question 5 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.
Question 6 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.
Question 7 is similar to my question 5 in Lesson 5 (Differentiation Rules).
Question 8 is a good definition of derivative problem as taught in Lesson 4.
Question 9 is a good run through of the differentiation rules as taught in Lesson 5.
Question 10 is trig limits as taught in question 16 of my Lesson 2. However, to solve this one, you need to know some trig identities. Click this link:
