Calculus I Honours Supplement

Upon completion of MATH 1121 the student should be able to:

• State the precise (epsilon-delta) definition of a limit
• Apply the (epsilon-delta) definition of a limit to evaluate limits of linear and quadratic 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 greatest integer function, 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
• Derive Newton’s Method and use it to solve problems