Course Title: Calculus II
Level: 1
Semester: II
No. of Credits: 3
Pre-requisite(s): Units 1 and 2 of CAPE Pure Mathematics or equivalent.
Course Rationale
Mathematics is a powerful tool used for solving practical problems and is a highly creative field of study, combining logic and precision with intuition and imagination. The ability to employ mathematical reasoning is a fundamental skill for any well-educated individual in the pure and applied sciences. This introductory level course is part of a sequence of courses designed to provide a solid foundation for students interested in further studies in Mathematics and also to give students in the Sciences the mathematical tools necessary for their work. It is a prerequisite for students who intend to take advanced level courses in Mathematics or statistics. It would also be very useful for students majoring in Physics, Chemistry and Biology.
This course is the second of a sequence of three Calculus courses. Calculus is one of the major tools used in applications of Mathematics. In addition, reflections on the foundations of Calculus form the starting point of the very large area of modern Mathematics known as Analysis. The two Level I Calculus courses build on work done in the Sixth form in calculus and introduce the student to greater rigour in their studies as well as to new topics in Calculus. This greater rigour forms an important bridge to later more abstract work in Analysis.
Course Description
This course will treat the limits, continuity and differentiability of a function of a single variable (previously met in Calculus I) from a more rigorous point of view. Double integrals are introduced and methods for their evaluation are considered. Prior knowledge of the content of Calculus I will be assumed.
Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through assigned problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and in the Help-Desk. Assessment will be continual and will be based on the assignments and in-course tests followed by a final examination based on the whole course.
The Help-Desk will be manned by teaching assistants who will give assistance (outside of the formal tutorials) to students on a one-on-one basis or in small groups
How to Study For This Course
“There is no royal road to Geometry” Response by Euclid to King Ptolemy’s request for a quick introduction to geometry.
Mathematics is not a spectator sport. In order to develop understanding and problem solving skills it is essential that students do many exercises and problems, including many problems additional to those set in assignments. This is the best way to learn the material properly and also to revise. One cannot learn Calculus by simply reading the notes given in class and looking over worked examples.
Students majoring in Mathematics and Actuarial Science will need to know and understand this material thoroughly and will use it in their future courses. It must become firmly entrenched in long-term memory, not something to be forgotten the day after the final examination. Again, the best way to do this is through a great deal of practice in solving problems. In Math 1152 we begin a more rigorous approach to calculus so it is particularly important that students follow-up on course lectures by reading through the definitions and proofs given in class. A good way to do this is by careful revision of lecture notes, and consultation with lecturers and tutors whenever necessary.
Do not forget that you have a text-book. Read it! You will get a second perspective on the material done in your lectures. You will also develop the ability to study and learn independently, instead of looking to your lecturer to be the sole source of new mathematical knowledge. Your textbook also contains many exercises, problems and examples. You should try them. Do not wait for your lecturer to tell you what to do. He might not!
The Department operates a help desk. This provides one-on-one assistance for students by Teaching Assistants (who run your tutorials). You should take advantage of this facility. Check MyeLearning or the Departmental Office for information about this.
It should be stressed that relying solely on the solution of past papers is not the ideal way to study for coursework or final examinations. For this purpose, more time should be dedicated to doing additional problems and to revising the assignment problems that have been discussed during tutorials, as well as the proofs and examples given during lectures.
Learning Outcomes
Upon successful completion of this course, students will be able to:
- Define and give examples of neighbourhoods of points.
- Prove the limit of a function using the definition of a limit.
- Evaluate limits, including limits at infinity.
- Examine the behavior of curves with asymptotes.
- Determine if a function, or its inverse, is continuous at a point.
- State and use the Intermediate and Extreme Value Theorems.
- Use the definition of differentiability to determine if a function is differentiable at a point.
- Prove basic theorems concerning derivatives.
- Use Rolle’s Theorem and the Mean Value Theorems.
- Evaluate indeterminate forms using L’Hospital’s Rule.
- Evaluate double integrals.
Course Content
- Intervals, neighbourhoods and bounds of a function (of a single variable).
- The definition of a limit.
- Properties/theorems of limits, with associated proofs.
- Directed (left-hand and right-hand) limits.
- Continuity, (removable and essential discontinuities).
- Properties/theorems of continuous functions.
- Intermediate Value Theorem.
- The Squeeze Theorem.
- The derivative of a function, (definition, differentiability continuity, left & right- hand derivatives).
- Rolle’s Theorem.
- Mean Value Theorem, (including Cauchy’s Mean Value Theorem).
- Evaluating indeterminate forms & using L’Hospital’s rule.
- Other indeterminate forms: , , , , .
- Reduction Formulae.
- Introduction to the Double Integral as a double sum.
- The Double integral as an iterated integral.
- Transformations in double integration.
Teaching Methodology
Lectures: Two (2) lectures each week (50 minutes each). Lecture notes will be made available to students via myeLearning.
Tutorial: One (1) tutorial weekly (50 minutes). During these tutorials, the solutions to assignments will be discussed (the focus will be on difficulties encountered), and extra problems may be suggested to the student for further practice. Students will be encouraged to participate by solving problems in class. Attendance at tutorial sessions is mandatory and will be monitored accordingly.
Assessment
Coursework: 50% consisting of two equally weighted in-course tests (40%) and assignments (10%)
Final Examination (one 2-hour written paper): 50%
You will be given one assignment per week (a total of 11), all of which will be graded. The final examination consists of essay type questions involving definitions, statements of theorems, proofs of major theorems and problems. The coursework examinations consist of short essay type questions and multiple choice questions which focus more on knowledge and understanding and solving simpler problems. The final examination will place greater emphasis on problem solving skills. The assignments consist of problems based on lectures that are covered in each week. These questions are compulsory. They are given out each week, as indicated in the course calendar.
Course Calendar
Week |
Lecture subjects |
Tutorials / Coursework examinations |
---|---|---|
1 |
Course Introduction and Overview The definition of a limit: Intervals, neighbourhoods and bounds of a function (of a single variable). The definition of a limit. |
None. |
2 |
Properties of limits: Properties/theorems of limits, with associated proofs; left and right hand limits; |
Assignment # 1 given. Quiz 1. |
3 |
Properties of continuous functions. Left and right continuity; Intermediate Value Theorem and the Squeeze Theorem. |
Assignment # 2 given. Assignment #1 due. |
4 |
Derivative of a function: Definition, differentiability continuity, left & right-hand derivatives. Rolle’s Theorem. Mean Value Theorem. |
Assignment # 3 given. Assignment #2 due. Quiz 2. |
5 |
Indeterminate forms: The use of L’Hopital’s rule to work out the indeterminate forms, & . Other indeterminate forms: , , , , . |
Assignment # 4 given. Assignment #3 due. |
6 |
Extreme Value Theorem. Asymptotes. |
Assignment # 5 given. Assignment # 4 due. Quiz 3. |
7 |
Length of a curve; reduction. |
Assignment # 6 given. Assignment #5 due. First coursework examination. |
8 |
Idea of a double integral as a sum |
Assignment # 7 given. Assignment #6 due. |
9 |
Evaluating double integrals via iterated integration. |
Assignment # 8 given. Assignment #7 due. Quiz 4. |
10 |
Transformations |
Assignment # 9 given. Assignment #8 due. |
11 |
Polar coordinates; more examples of double integrals and transformations |
Assignment # 10 given. Assignment #9 due. Quiz 5. |
12 |
Rigorous definition of Limit of a sequence of real numbers; limits of a sum, product and quotient; |
Assignment # 11 given. Assignment #10 due. |
13 |
Revision |
Assignment #11 due. Second coursework examination. |
Required Reading
- The Elements of Calculus-Harold Ramkissoon, Charles de Matas (2011).
- Lecture notes will be made available via myeLearning.
Recommended Texts:
- Single Variable Calculus-C.H. Edwards & D.E.Penny, sixth edition, Prentice Hall, ISBN 0-13-736331-1, (2000).
- Calculus by Frank Ayres and Elliott Mendelson (Schaum’s Outlines).
Other Reference Texts:
- Calculus – Anton Howard, fourth edition, John Wiley & Sons, ISBN 0-471-63631-2, (1998).
- Calculus with Analytical Geometry – Louis Leithold, fourth edition, Harper & Row , New York, ISBN -0 – 06-350401-4, (2000).
- Calculus by Serge Lang.