Course Description
This course is a supplement to MATH 1120-Calculus I with an emphasis on proving theorems from MATH 1120. Topics include the limit, continuity, differentiability, proof of differentiation rules, proof of the Mean Value Theorem and L’Hôpital’s rule.
Course Content
- Introduction to proof
- Properties of the real numbers
- Limits
- Continuity
- Differentiability
- Applications of differentiability
Learning Activities
Lectures, assignments, group work
Means of Assessment
Assessment will be in accordance with the Douglas College Evaluation Policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following
Tests: 0-70%
Assignments: 0-80%
Group work: 0-10%
Attendance: 0-5%
Final examination: 20-40%
Total: 100%
Learning Outcomes
Upon successful completion of the course, students should be able to:
- use the axioms of the real numbers to prove the Triangle Inequality and related inqualities.
- state the precise (epsilon-delta) definition of a limit.
- apply the (epsilon-delta) definition of a limit to evaluate limits of linear and power functions.
- state the precise (epsilon-delta) definition of continuity at a point.
- prove a function is continuous at a point using the precise definition of continuity at a point.
- prove that algebraic combinations of functions are continuous at a point.
- prove that functions are continuous on an interval.
- solve existence problems using the Intermediate Value Theorem.
- state the definition of the derivative.
- apply the concepts of the limit, continuity and differentiability to the absolute value function, the floor and ceiling functions, the Dirichlet function and the modified Dirichlet function.
- prove the linearity rules for differentiation using the definition of the derivative.
- prove the product rule for differentiation using the definition of the derivative.
- prove the quotient rule for differentiation using the definition of the derivative.
- prove the chain rule for differentiation using the definition of the derivative.
- prove the power rule for non-negative exponents, integer exponents, rational exponents and real number exponents.
- prove the differentiation formulas for trigonometric functions using the definition of the derivative.
- prove the differentiation rule for the inverse of a differentiable function.
- prove L’Hôpital’s rule for the case of “0/0”.
- prove Fermat’s Theorem, Rolle’s Theorem and the Mean Value Theorem.
- use the Mean Value Theorem to prove properties of differentiable functions such as, but not limited to, the test for monotonicity and the test for concavity.
Textbook Materials
Textbooks and materials are to be purchased by students. A list of required and textbooks and materials is provided for students at the beginning of the semester. Example texts may include:
Michael Spivak's Calculus (3rd edition or later).