Course Title: Calculus I
Level: 1
Semester: 1
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 those majoring in Physics, Chemistry and Biology.
This course is first one of a sequence of three Calculus courses. The Calculus is one of the major tools used in applications of Mathematics. In addition, reflections on the foundations of the 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 the Calculus. This greater rigour forms an important bridge to later more abstract work in Analysis.
Course Description
The course covers the basic ideas of the Calculus of one variable and introduces the differential calculus of several variables. The approach is intuitive and informal rather than rigorous. A sound knowledge of the Calculus modules of Units 1 and 2 of CAPE are assumed. There is considerable overlap with the ideas encountered in CAPE but there is greater emphasis on fundamental ideas involved in differentiation and the definite integral. The main results are derived but the approach is kept at a non-rigorous level.
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 or 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. It is particularly important that students follow-up on course lectures by reading through the definitions and proofs given in class. The best 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 textbook! Read it! It will give you a second perspective on the material in the course and will make you less dependent on your lecturer. Independent reading also develops your ability to study and learn on your own. Textbooks are also valuable sources of examples and problems.
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 revising the assignment problems that have been discussed during tutorials, as well as the proofs and examples given during lectures.
AIMS OF THE COURSE
- To introduce the ideas of limit of a function andof a sequence.
- To apply the informal notion of a limit to define and determine continuity.
- To apply the idea of a limit to obtain the derivative of well-known functions.
- To apply the idea of limit to define and determine definite integrals.
- To use various techniques to evaluate integrals.
- To define the sum of an infinite series and to test series for convergence.
- To introduce the differential calculus of several variables.
Learning Outcomes
Upon successful completion of this course, students will be able to:
- Define the ideas of a limit of a sequence and limit of a function.
- Evaluate limits of simple functions, including trigonometric functions
- Determine whether a function is continuous at a point.
- Use the Intermediate Value Theorem to find the roots of equations.
- Obtain the derivatives of functions using the definition.
- Calculate derivatives of functions using the chain rule.
- Calculate the Riemann integral of simple functions from the definition of the integral as a sum.
- State and use the Fundamental Theorem of the Calculus.
- Use the main techniques of integration, including trigonometric substitutions.
- Define what is meant by a convergent sequence and a convergent series.
- Test series for convergence.
- Find the Maclaurin and Taylor series expansions of functions.
- Determine the partial derivatives of functions of several variables.
- Find the maxima and minima of functions of several variables.
Course Content
- Limit of a Sequence of real numbers treated intuitively. Sum, product and quotient of convergent sequences without proof.
- Partial sum of a series real numbers. Definition of a convergent series and examples of convergent and divergent series.
- The Comparison and ratio tests for convergence of a series.
- Limits of a function. Basic properties of limits without proof. Limit of sin(x)/x as x tends to zero.
- Limit as x tends to infinity.
- Evaluating the limits of functions.
- Definition of Continuity at a point. Examples of dis-continuous functions.
- The Intermediate value Theorem and its use to find roots of equations.
- Definition of the derivative as the limit, as h→0 of (f(x+h)-f(x))/h.
- Calculating the derivative of simple functions using the definition.
- Derivation of the derivative of the sum, product and quotient of functions.
- Leibniz’s formula.
- The chain rule.
- Hyperbolic functions.
- The definite integral as the limit of a sum.
- Evaluating the (Riemann) integral of simple functions from the definition.
- Statement and use of the Fundamental Theorem of the Calculus.
- Evaluation of integrals by standard techniques.
- Length of a curve.
- Functions of two variables and their graphs. Functions of several variables.
- Definition and calculation of the partial derivative of a function of several variables.
- Maxima and minima of functions of two variables.
Teaching Methodology
Lectures: Two (2) lectures each week (50 minutes each). Lecture notes will be made available to students via myeLearning. Assignments will be given out every week. Solutions to assignments will be posted in-line and some may be discussed in tutorials.
The traditional lecture is still the main mode of instruction but most lecturers make use of multi-media resources and of alternative teaching strategies.
Tutorial: One (1) tutorial weekly (50 minutes). During these tutorials, difficulties in the assignments will be discussed, and extra problems may be suggested to the student for further practice. Students will be encouraged to participate by solving problems in class and by working in small groups. Attendance at tutorial sessions is mandatory and will be monitored accordingly.
Help-Desk Students are encouraged to make use of the help-desk which is manned by Teaching Assistants and demonstrators. This provides one-on-one assistance.
Assessment
Coursework: 50%, comprised 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 quizzes and coursework examinations consist of multiple choice questions and shorter essay type 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 normally given out each week, as indicated in the course calendar.
Course Calendar
Week |
Lecture subjects |
Tutorials / Coursework examinations |
---|---|---|
1 |
Introduction/Course Overview Limit of a function; properties of limits; limit of sin(x)/x as x tends to zero; limit as x tends to infinity. |
None |
2 |
Continuity; examples of dis-continuous functions; the Intermediate Value Theorem; finding roots of equations. |
Assignment # 1 given; Quiz 1. |
3 |
Definition of derivative; derivatives of sum, product and quotient; derivative of simple functions from the definition. |
Assignment # 2 given; Assignment #1 due |
4 |
The chain rule; hyperbolic functions; sequences; limit of a sequence; convergent and divergent series. |
Assignment # 3 given: Assignment #2 due. Quiz 2. |
5 |
Applications of differentiation: Taylor and Maclaurin series for polynomial approximations. |
Assignment # 4 given; Assignment #3 due |
6 |
The Definite (Riemann) integral as the limit of a sum; integration of simple functions from the definition; |
Assignment # 5 given; Assignment #4 due. Quiz 3. |
7 |
The Fundamental Theorem of the Calculus; integration by substitution (including trig. Substitution). |
Assignment # 6 given; Assignment #5 due; First coursework examination |
8 |
Integration by parts; Length of a curve; |
Assignment # 7 given; Assignment #6 due |
9 |
Functions of two and their graphs; functions of three and more variables; continuity of functions of two variables. |
Assignment # 8 given; Assignment #7 due; Quiz 4. |
10 |
Partial derivatives; the total differential |
Assignment # 9 given; Assignment #8 due |
11 |
Maxima and minima of functions of two variables |
Assignment # 10 given; Assignment #9 due; Quiz 5. |
12 |
Tests for convergence of infinite series. |
Assignment # 11 given; Assignment #10 due; Second Coursework examination. |
13 |
Revision |
Assignment #11 due |
Required Reading
- The Elements of Calculus by Harold Ramkissoon and Charles de Matas.
- Calculus 5^{th} ed. Frank Ayres and Elliott Mendelson (Schaum’s Outlines).
Reference Texts
- Single Variable Calculus-C.H. Edwards & D.E.Penny, sixth edition, Prentice Hall, ISBN 0-13-736331-1, (2000).
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