Pre-Calculus and the Derivative
- Course ID:MATH 445/446
- Teachers:Michael Moynihan
Description and Objectives
- A completion of pre-calculus and an introduction to limits and differential calculus
- Special attention will be given to helping students master the art of graphing first using non-calculus based analytical tools and later with the help of the derivative.
- By carefully working through difficult systems and mathematical problems, students will continue to develop their ability to think logically and clearly, with a mind well trained to make reasonable distinctions. Mathematics has always been important for developing this type of mental acumen, the reason why its study has traditionally been viewed as essential to developing the reasoning ability necessary for a later study of philosophy.
Outline of Topics Covered:
- Chapter P – Prerequisites. Review of material from earlier math courses including number theory, exponents and radicals, polynomials and factoring, solving equations, and basic graphing techniques
- Chapter 1 – Functions and Their Graphs. Review of function theory and a continuation of graphing from a numeric and analytic perspective including a foundational study of inverse functions
- Chapter 2 – Polynomial and Rational Functions. Quadratic and higher order polynomial functions analyzed with the help of factoring by synthetic division and the use of the chart of signs; a more in depth understanding of real and complex zeros; graphing rational functions with special attention to asymptotes (vertical, horizontal and slant), zeros, and holes (not in the text but covered anyway)
- Chapter 3 – Exponential and Logarithmic Functions. Graphing exponential and logarithmic functions; solving exponential and logarithmic equations; an introduction to natural logs and the transcendental irrational number e
- Chapters 4-6 – Trigonometry. A comprehensive study of trigonometry starting from the fundamental trigonometric relationships in triangles, progressing to the unit circle, and finally a full analytic and graphical treatment of trigonometric functions; fundamental identities form the basis of trigonometric proofs
- Chapter 7 – Systems of Equations and Inequalities. Graphing and solving systems of equations and inequalities from relatively simple linear ones to complex and multifaceted systems involving diverse types of functions and equations; the focus is on developing the art of approaching systems from both an algebraic perspective (substitution and elimination) and a graphical perspective at the same time
- Chapter 9 – Sequences. A study of sequences and series in general and a comprehensive treatment of arithmetic and geometric sequences; summation notation and the concept of the infinite series will be examined
- Limits – Presented as the y-value that a function approaches as x gets infinitely close to a given value, limits are explored from a graphical and algebraic perspective. Limits are used to better understand asymptotes (introduced in chapter 2) and holes.
- Differential Calculus – The starting point is a comprehensive study of the development of the derivative from the slope of the tangent line problem and how limits are essential to this great mathematical insight. Techniques for differentiation are studied and the derivative is used to help graph many functions previously studied in the course. Topics include increasing and decreasing functions; relative extrema; concavity and the second derivative; points of inflection; position, velocity and acceleration; and a more complete understanding of the number e and its use in forming a function that is its own derivative.
- Precalculus, 4th edition by Larson and Hostetler. ISBN 0669417416
- Additional materials presented in handout form for limits and differential calculus
- Mastery of mathematics on this level requires disciplined and focused effort over an extended period of time. Successful students, in addition to their study in class, will spend time every evening struggling to complete homework problems correctly. The majority of the problems assigned are odd numbered problems with answers in the back of the book. Students should always check to make sure they are finding their mistakes and doing the problems correctly. Homework and class work problems should be looked at as a means to study the material and grow in mastery.
- Homework and notebooks are checked for completion. Frequent quizzes with problems similar to homework problems ensure that students are mastering the homework problems. Tests are given approximately every two weeks and are more comprehensive and challenging than quizzes. Each test will have three sections: an initial “Did you study?” section where simple recall of memorized facts is required, a second and longer “Can you do math?” section where students work through relatively straight forward problems like those seen in the homework, and a final “Do you understand?” section with more challenging problems requiring mathematical reasoning. Points earned for quizzes and tests provide each student’s numerical quarter average, which in turn, may be modified slightly to reflect efforts on homework and class work.
- Students must have a non-graphing scientific calculator. Graphing calculators are not allowed for this course but will be used from time to time in AP Calculus next year. Students should bring their calculator, notebook, and textbook to class every day.
- Successful students will promptly memorize the things which must be memorized (formulas, theorems, certain relationship and facts) so that their main focus is on correctly doing the assigned problems. Excellence will be achieved by developing a spirit of contemplative study, pondering the relationships inherent in the material. Some of the better students are held back from real excellence by not slowing down from time to time to ponder the order, connectivity, and elegance in the field of mathematics.
- I am available for help outside of class, during most lunch periods and after school when I do not have meetings. I encourage parents to contact me with any questions or concerns either by email or phone.
MATH 445-446 Pre-Calculus and the Derivative
Mr. Michael Moynihan
Spend time reading through chapter P (stands for “prerequisites”) in your text, Precalculus 4th edition by Larson and Hostetler (ISBN 0669417416). These pages are available here: 1300_001 (1). This material should mostly be a review of what you have already covered in previous math courses. As you go through this material, do at least 150 problems from the text. While it is up to you what problems you choose, you will benefit most from this assignment if you choose problems that reinforce areas that need improvement. In other words, if you choose easy problems to simply complete the assignment, you will not benefit as much. You should choose problems that help solidify your foundation.
During the first part of the year we will cover this material quickly. If you do this assignment well, you will find the first part of the course easier. If you come to class without a strong foundation in the material this summer assignment covers, the beginning of the year will be difficult.
I will collect your summer assignment on the first day of school and grade it based on completion and effort shown (students who show their work and choose diverse problems will receive higher marks).