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Student Learning Outcomes -

• Complete applied real world problem projects with solutions and relevant explanations, accompanied with the use of mathematical typesetting software.
• Use formal definitions and theorems with mathematical proof techniques to prove limits, derivative values, and relevant theorems.

Description -

An honors seminar linked to MATH 1AH. In this course, students will explore a multitude of advanced problems from the calculus I honors course, including proofs of limit laws, differentiation rules, and corresponding theorems concerning the behavior of differentiable functions. As the calculus I honors course will require students to submit typed technical solutions to applied problems, this seminar will support students in learning how to use mathematical typesetting software. Best practices for mathematical writing will also be discussed.

Course Objectives -

1. State and prove limits.
2. State and prove derivatives.
3. State and prove theorems.
4. Demonstrate an understanding of applications of the derivative.

Special Facilities and/or Equipment -

1. Access to graphing technology, such as a graphing calculator or graphing software.
2. Access to Mathematical Typing Software.
3. When taught via 17³Ô¹Ï Global Access, on-going access to computer with email software and hardware; email address.

 

Course Content (Body of knowledge) -

1. State and prove limits.
1. Epsilon-delta proofs of limits
2. Proofs of limit laws
3. Proofs involving continuity
4. L'Hospital's rule
2. State and prove derivatives.
1. Epsilon-delta proofs
1. Derivative values
2. Second derivative values
2. Proofs of derivative rules
1. Power rule
2. Product rule
3. Derivatives of logarithmic functions
4. Derivatives of trigonometric functions
5. Derivatives of inverse functions
3. State and prove theorems.
1. Mean Value Theorem
1. Rolle's Theorem
2. Intermediate Value Theorem
3. Extreme Value Theorem
4. The Squeeze Theorem
4. Demonstrate an understanding of applications of the derivative.
1. Related rates
2. Optimization
3. Linear approximations
4. Differentials

Methods of Evaluation -

1. Typed formal proofs
2. Special applied projects
3. In class presentations

Representative Text(s) -

1. Briggs W., L. Cochran, and B. Gillett. Calculus Early Transcendentals. 2nd ed. Boston: Pearson, 2015.
2. Instructor-generated materials, such as excerpts from:
1. Trench, William F. Introduction to Real Analysis. Free Edition Open Textbook Online, previously published by Pearson Education.
2. Lay, Steven R. Analysis With an Introduction to Proof. 5th ed. Pearson Education, Inc., 2014.

 

Disciplines -

Method of Instruction -

1. Lecture
2. Discussion
3. Cooperative learning projects

Lab Content -

Types and/or Examples of Required Reading, Writing and Outside of Class Assignments -

1. Homework Problems: Homework problems covering the subject matter from the text. Honors students will be assigned more of the challenging problems from the text on a regular basis.
2. Special Applied Projects: At least one applied real world project which will be typed using appropriate math typing software. Projects will also be presented in class.
3. Typed proofs: Formal proofs which will be typed and accompanied with math typing software.