Gary Punchard
gary@punchard.com

What is Calculus:
Calculus was developed out of a need to understand continuously changing quantities. Newton, for example, was trying to understand the effect of gravity which causes falling objects to constantly accelerate. The speed of an object increases constantly every split second as it falls. How can one, for example, determine the speed of a falling object at a frozen instant in time, such as its speed when it strikes the ground? No mathematics prior to Newton and Leibnitz's time could answer such a question, which appeared to amount to the impossibility of dividing zero by zero. The solution to this type of issue came to be known as the derivative. Derivatives are slopes of particular lines called tangent lines, and the reader may recall that slope of a line is a concept from Descartes' graphing.

Differential calculus is one side of calculus, the part concerned with continuous change and its applications. By understanding derivatives the student has at his or her disposal a very powerful tool for understanding the behavior of mathematical functions. Importantly, this allows us to optimize functions, which means to find their maximum or minimum values, as well as to determine other valuable qualities describing functions. Real-world applications are endless, but some examples are maximizing profit, minimizing stress, maximizing efficiency, minimizing cost, finding the point of diminishing returns, and determining velocity and acceleration.

The other primary side of calculus is integral calculus. Integration is a process which, simplistically, resembles the reverse of differentiation. This amounts to efficiently adding infinitely many infinitely small numbers. This allows us, in theory, to find the area of any planar geometric shape, or the volume of any geometric solid. But the applications of integration, like differentiation, are also quite extensive.

What is Expected:
The student is expected to come to class ready to learn.  Homework is not assigned on a daily basis; instead, the student will be keeping a journal in class with notes and class work.  This journal will be used when taking quizzes and for the semester final; it will not be used for unit tests.  At the end of each quarter, the journal will be graded.

What is Needed:
The student will need a 3 ring binder (with name on binding) with loose leaf lined paper and writing utensil. The binder will be left in the classroom unless the student needs to make up work or study for a test.  The student must have journal and writing utensil ready at the beginning of each class.  Occasional quizzes will be given that require the journal.  If a student is in possession of someone else’s journal, that student will receive an automatic zero on the next quiz given; an exception will be made for copying notes as long as the owner of the journal has given permission. The student will be issued a textbook or text on computer disk to take home. These must be returned in like condition.

Grading:

• 60% Tests (Three per Quarter)
• 20% Quizzes (10-15 per Quarter)
• 20% Journal (1 per Quarter).

Overall Grading:

• 40% First (Third) Quarter
• 40% Second (Fourth) Quarter
• 20% Semester Final

Quarter 1

Greek Alphabet

Trigonometry Review

• T.1 Defining an Angle
• T.2 Conversion Between Degrees and Radians
• T.3 Defining the Trig Functions
• T.4 Trig Values for Special Acute Angles
• T.5 Trig Functions of Angles Lying on the Axes
• T.6 Trig Functions of Any Angle - The Generalization
• T.7 Graphs of the Trigonometric Functions
• T.8 Identities Involving Trigonometric Functions

Chapter 1. Preparation for Calculus

• 1.1 Graphs and Models
• 1.2 Linear Models and Rates of Change
• 1.3 Functions and Their Graphs
• 1.4 Fitting Models to Data
• 1.5 Inverse Functions
• 1.6 Exponential and Logarithmic Functions

Quarter 2

Chapter 2. Limits and Their Properties

• 2.1 A Preview of Calculus
• 2.2 Finding Limits Graphically and Numerically
• 2.3 Evaluating Limits Analytically
• 2.4 Continuity and One-Sided Limits
• 2.5 Infinite Limits
• Section Project: Graphs and Limits of Trigonometric Functions

Chapter 3. Differentiation

• 3.1 The Derivative and the Tangent Line Problem
• 3.2 Basic Differentiation Rules and Rates of Change
• 3.3 The Product and Quotient Rules and Higher-Order Derivatives
• 3.4 The Chain Rule

Quarter 3

• 3.5 Implicit Differentiation
• Section Project: Optical Illusions
• 3.6 Derivatives of Inverse Functions
• 3.7 Related Rates
• 3.8 Newton's Method

Chapter 4. Applications of Differentiation

• 4.1 Extrema on an Interval
• 4.2 Rolle's Theorem and the Mean Value Theorem
• 4.3 Increasing and Decreasing Functions and the First Derivative Test
• Section Project: Rainbows
• 4.4 Concavity and the Second Derivative Test
• 4.5 Limits at Infinity
• 4.6 A Summary of Curve Sketching
• 4.7 Optimization Problems
• Section Project: Connecticut River
• 4.8 Differentials

Quarter 4

Chapter 5. Integration

• 5.1 Antiderivatives and Indefinite Integration
• 5.2 Area
• 5.3 Riemann Sums and Definite Integrals
• 5.4 The Fundamental Theorem of Calculus
• Section Project: Demonstrating the Fundamental Theorem
• 5.5 Integration by Substitution
• 5.6 Numerical Integration
• 5.7 The Natural Logarithmic Function: Integration
• 5.8 Inverse Trigonometric Functions: Integration
• 5.9 Hyperbolic Functions
• Section Project: St. Louis Arch