PMAT 12253: Theory of Calculus

Course code  :  PMAT 12253

Title               :  Theory of Calculus

Pre-requisites :  PMAT 11223

Learning Outcomes:

On successful completion of the course, the student should be able to;

1. classify a real number as a natural, whole, integer, rational, or irrational number and demonstrate knowledge
of the axiomatic description of the field of real numbers and prove theorems from the given set of axioms
2. evaluate limits given analytic, graphical, numerical function information and describe in simple language the
statements of limit laws and use these laws to evaluate limits and state the definition of continuity and use
the definition to ascertain the continuity or dis-continuity of a function at a point
3. state the limit definition of derivative of a function at a point and use the limit definition to calculate a
derivative or identify where the derivative fails to exist at a point
4. apply the Chain Rule to find the derivative of a composition of functions and interpret both continuous and
differentiable functions geometrically and analytically and apply Rolle's Theorem, the Mean Value Theorem
5. explain indeterminant forms and use L’Hopital’s rule to evaluate limits involving indeterminant forms
6. describe upper and lower Darboux sums, Riemann sum and compare them and write the statement of the
Fundamental Theorems of Calculus and explain what the theorems say about definite integrals
7. visualize and sketch the surface generated by revolving a graph of a function about an axis of evolution and
calculate the volume of a solid and volume of a solid revolution by using disk, washer and cylindrical
shells methods
8. classify improper integrals and determine the integrals, identify the types of improper integrals and rewrite
them as proper integrals with a limits and use the words convergent and divergent to describe an
improper integral.

 Course Contents:

Real Numbers: 

• Field Structure, Ordered fields and their properties, Open and Closed sets, Maximum, Minimum, Supremum and Infimum of a set, Completeness axioms, Archimedean Property, Denseness of subsets of real numbers

Real Valued Functions of a Real Variable: 

• Review of Polynomial, Rational, Algebraic, Trigonometric, exponential and Logarithmic Functions, Composite Functions, Piece-wise Functions

Functions and Limits: 

• Limits of Functions, Left and Right-hand limits, Squeeze theorem, Continuous Functions, Asymptotes and limits involving infinity

Derivative and Applications: 

• Derivative of a function, Chain rule, Logarithmic and Implicit differentiation, Higher order derivatives, Rolle’s Theorem, Mean Value Theorem

Indeterminate Forms: 

• L’hospital Rule. Integrals: Darboux and Riemann Integrations, Fundamental Theorem of Calculus, Improper Integrals (First &
  second kind)

Applications of Integrals: 

• Volumes and Solid revolutions, Volumes by Cylindrical Shells, Mean Value Theorem
  for Integrals, Arc Length, Area of a Surface of Revolution

 Method of Teaching and Learning : A combination of lectures and tutorial discussions.

 Assessment : Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart, J. (9th Ed., 2020). Calculus Early Transcendentals, Cengage Learning, Inc.
2. Larson, R. & Edwards, B.H. (11th Ed., 2018). Calculus, Brooks/Cole, Cengage Learning
3. Ross, K.S. (2nd Ed., 2015). Elementary Analysis: The Theory of Calculus, Springer.
4. Hass, J., Heil, C. & Weir, M.D. (14th Ed., 2017). Thomas' Calculus: Early Transcendentals, Single
Variable, Pearson