Course Information
I generally teach Honors Calculus I, as opposed to AP Calc. For me, what this means is that the course emphasizes depth of content including proofs, reasoning, concepts and developing the mathematics in addition to the actual problem solving. I feel as though AP Calculus tends to be surface level knowledge, but with heavy emphasis on problem solving and drilling examples. In my opinion, this trains people to follow instructions without knowing any real math. I prefer to teach the understanding, background and development with practice problems as a secondary focus. However, I hope there are enough practice exercises provided here to ensure that anyone has as many examples as they need and more.
- Course Overview – This is what I handout on the first day of class. It contains a quick summary of the course, requirements, and procedures for an 11th grade course.
- Yearly Plan – This is how I break down my units in Calculus I and the number of days I reserve for specific topics, based on seven years of experience. NOTE: I add in “Extra” days for each unit. These serve both as a buffer in case more time is needed and as a break from the normal math. They are generally designed to supplement a topic in the class, provide historical background, open the students to thinking differently, a chance to read an original text, or something else.
- Formula Sheets
Required Knowledge: Students engaging with Differential Calculus should be familar with the following concepts covered in the previous courses of Algebra and Trigonometry:
- Functions – Evaluating, Graphing, Defining
- Solving Equations
- Basic Geometry (Shapes and Formulas, not proofs)
- Quadratic Equations and the Quadratic Formula
- Absolute Values
- Fractions and Decimals – Converting between them, Simplifying Fractions, Performing Arithmetic Operations on them
- Exponents, Roots, Logarithms and the relationship between them
- Trigonometry – Six main functions, Definitions, Unit Circle, Inverse Trigonometry, Basic Trigonometric Identities, Understanding of Radians and Degrees
- Inverse Functions – Identifying, Graphing, and Solving for Inverses
- Linear Equations – Point Slope Form, Slope Intercept Form, How to find Slope, Graphing
- How to use a Calculator
To practice these concepts and more, see previous Helpful Tools for Calculus.
Table of Contents
Chapter 3: Limits – An Introduction to the Concepts of Rate of Change, Limits, leading to the Formal Definiton of a Limit
- The Average Rate of Change
- The Instantaneous Rate of Change
- The Limit
- Limit Laws
- Limits, Graphically
- Continuity
- Evaluating Limits with Algebra
- Evaluating Trigonometric Limits
- The Squeeze Theorem
- Infinite Limits
- The Intermediate Value Theorem
- The Formal Definition of a Limit (Epsilon-Delta Proof)
Chapter 4: Derivatives – The Definition of a Derivative, the Creation and use of Derivative Rules and Methods of Evaluating a Derivative
- The Limit Definition of a Derivative
- The Derivative as a Function
- Basic Derivative Rules (Power, Constant, Sum)
- Estimating Derivatives Graphically
- Trigonometric Derivatives
- Product and Quotient Rules
- Chain Rule
- Multiple Derivatives
- Exponential and Logarithmic Derivatives
- Derivatives of Inverse Functions
- Derivatives of Hyperbolic Functions
- Implicit Differentiation
- Derivatives of Parametric Equations
- Derivatives of Polar Equations
Chapter 5: Applications of Derivatives – Using Differential Calculus to solve word problems and identify relationships or trends in Functions.
- Related Rates
- L’Hôpital’s Rule
- Extreme Values
- The Mean Value Theorem
- First and Second Derivative Tests
- Graphing a Function with Derivatives
- Optimization
Chapter 6: Introduction to Integral Calculus – Antiderivatives, Summation, and the Fundamental Theorem of Calculus
- Summation
- Using Rectangles to Estimate the Area under a Curve
- Using Infinitely Many Rectangles
- The Antiderivative
- The Fundemental Theorems of Calculus