Course Code:                    MATH 1142

Course Title:                     Calculus I

Level:                                  1

Semester:                            1

No. of Credits:                      3

Pre-requisites(s):                 Two units (1 & 2) of CAPE Pure Mathematics or MATH 0100 and MATH 0110; 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 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

 

This course will cover the basic ideas and techniques of Calculus.  The approach is more intuitive and informal than in succeeding courses. It will look at some of the well-known and most useful functions, representation of a point both in Cartesian and Polar co-ordinates,  techniques of differentiation, with graphical and function approximation applications.  Various methods of indefinite integration with some geometric applications will also be done. Methods for solving first and second order ordinary differential equations are also included.

 Prior knowledge of CAPE Advanced Level Proficiency in Mathematics (or its equivalent) will be assumed. 

 

All lectures, assignments, handouts, and review materials are available online through myeLearning to all students. Blended leaning techniques will be employed. Lectures will be supplemented with laboratory work and group discussions.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the assignments and in-course tests followed by a comprehensive final examination.

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.

 

Learning Outcomes

 

Upon successful completion of this course, students will be able to:

 

  • Represent a point in Cartesian or Polar co-ordinates.
  • Differentiate functions using the product rule, quotient rule, chain rule, Leibnitz’s Formula and using implicit differentiation.
  • Differentiate a function defined in terms of a parameter.
  • Use differentiation to determine relative extrema.
  • Sketch curves.
  • Use Taylor and Maclaurin series for polynomial approximations.
  • Integrate expressions involving standard functions using basic techniques including substitution, integration by parts and by partial fractions.
  • Apply integration to determine the length of curves and areas of regions.
  • Solve first order differential equations, using various methods.
  • Solve second order ordinary linear homogeneous and non-homogeneous differential equations with constant coefficients.