Calculus I Honours Supplement

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.