2015 - 2016 Spring Semseter MATH 1350 Survey of Calculus
Study Guides for the Suggested Exercises
- Exercises: Section 2.1 # 15, 16, 21, 23
- Remarks: In these exercises, the student is asked to find function values and limits for functions f and g that are given by graphs (not given by formulas). One could visualize the idea of these problems with an arrow diagram:
Graph ==> Description of Limit Behavior.
- Relevant Reading: Section 2.1 pages 95 - 98
- Useful Book Examples: Section 2.1 Example 4 (and corresponding Matched Problem)
- Useful Lecture Notes: Class Meeting #1
- Useful Class Drills: Class Drill #1
- Exercises: Section 2.1 # 47, 49
- Remarks: In these exercises, the student is given a description of Limit Behavior, and is asked to draw a graph that has that behavior. One could visualize the idea of these problems with an arrow diagram:
Description of Limit Behavior ==> Graph.
- Relevant Reading: Section 2.1 pages 95 - 98 contain the general exposition about Limits, but there is no discussion of problems such as these.
- Useful Book Examples: There are no similar book examples.
- Useful Lecture Notes: Class Meeting #1 lecture notes will be very important, because there are no similar book examples!
- Exercises: Section 2.1 # 33, 35, 37
- Remarks: In these exercises, the student is asked to find function values and limits for functions that are given by graphs formulas. The idea is to use Theorem 2 (Properties of Limits) and Theorem 3 (Limits of Polynomial and Rational Functions). The main point of the "solution" of problems like these is to show each application of a Theorem on its own line, with the particular Theorem cited by number (such as Theorem 2.6). (See book examples and class notes for models.) In all three of these basic exercises, the value of the limit at x=c ends up equaling the function value at x=c. But again, the limit is to be computed by using the Theorems and citing them in the written solution, not simply by just substituting in x=c.
- Relevant Reading: Section 2.1 pages 95 - 101
- Useful Book Examples: Section 2.1 Examples 5,6 (and corresponding Matched Problems)
- Useful Lecture Notes: Class Meeting #2
- Useful Reference Pages: Reference 2: Facts about Limits from Section 2.1 on page 2 of the Course Packet.
- Exercises: Section 2.1 # 51, 53, 57, 91
- Remarks: In these exercises, the function is a piecewise defined function. That is, the domain of the function is broken up into two or more pieces, and the formula that describes the function is different on each piece. In problems #51 and #53, the piecewise definition is explicit: The pieces of the domain are presented clearly, and for each piece of the domain, the formula for the function is given clearly. Problem #57 is not presented as a piecewise-defined function: the formula is just a single expression. But notice that the expression involves the Absolute Value Function. In order to work with the Absolute Value fFnction, you will have to realize that it, too, is a piecewise-defined function. (The piecewise definition of the Absolute Value Function is given just before Example #3 on page 97. Problem #91, about Telephone Rates, asks the student to come up with a piecewise definition of the charge function F, and then graph it and answer questions about its limit behavior.
- Relevant Reading: Section 2.1 pages 95 - 101
- Useful Book Examples: Section 2.1 Examples 3, 7, 8B (and corresponding Matched Problems)
- Useful Lecture Notes: Not covered in lecture: be sure to study the book!
- Exercises: Section 2.1 # 61, 63, 77, 79
- Remarks: In these exercises, the function is a rational function. A crucial concept is the distinction between the function value at x=c and the limit at x=c. In some problems, the function value does not exist at x=c but the limit does exist at x=c. But in other problems, niether the function value nor the limit exists at x=c. Perhaps the most important concept of the first month of the course is the idea that sometimes one cannot cancel expressions when computing function values (if the expressions result in 0/0), but can cancel those same expressions when computing limits. The book does a lousy job with this concept in Section 2.1. In Example 8, the book says that "...Algebraic simplification is often useful ...", but this is very vague and does not address the underlying issue. The "Conceptual Insight" at the top of page 103 attempts to explain the concept. But nothing is mentioned about the fact that when one is actually computing a function value (not computing a limit), one is not allowed to cancel. In lecture, I will be very explicit about the important underlying concept. And I will be very clear in lectures about how I expect these sorts of problems to be written. It is important that in your solutions, you explain the important underlying concept clearly. (Again, note that the book solutions do not explain the underlying concept at all. So I will be expecting your written solutions to be more thorough than the solutions presented in the book!)
- Relevant Reading: Section 2.1 pages 95 - 104
- Useful Book Examples: Section 2.1 Examples 8A, 9 (and corresponding Matched Problems)
- Useful Lecture Notes: Class Meeting #2
- Useful Reference Pages: Reference 2: Facts about Limits from Section 2.1 on page 2 of the Course Packet.
- Exercises: Section 2.1 # 81, 83
- Remarks: In these exercises, the student is given a formula for a function f and must build a Difference Quotient for f and then take a limit of that Difference Quotient. The calculations are difficult, and they seem rather random: the book mentions that Difference Quotients are "important", but does not explain why they are important. It turns out that Difference Quotients are central to the upcoming definition of the Derivative in Section 2.4.
- Relevant Reading: Section 2.1 pages 95 - 104
- Useful Book Examples: Section 2.1 Example 10 (and corresponding Matched Problem)
- Useful Lecture Notes: Class Meeting #2
- Exercises: Section 2.1 # 67,68,69,70,71,72
- Remarks: These exercises require no computation, but to answer them will require that you read Section 2.1 very carefully. The clues that you need to answer these six questions are scattered throughout the reading.
- Relevant Reading: Section 2.1 pages 95 - 105
- Useful Book Examples: Section 2.1 All 10 Examples (and corresponding Matched Problems)
- Useful Lecture Notes: Class Meetings #1 and #2
(page maintained by Mark Barsamian, last updated Jan, 2016