Course Information
I generally teach Honors Calculus II, 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 a 12th grade course.
- Yearly Plan – This is how I break down my units in Calculus II and the number of days I reserve for specific topics, based on five 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 Integral Calculus should be familar with the following concepts covered in the previous course of Calculus I (Differential Calculus) as well as Algebra and Trigonometry:
- Limits
- Definition of a Derivative
- Derivative Rules
- Applications of Derivatives
- L’Hôpital’s Rule
- Summation
Table of Contents
Chapter 5: The Integral – Defining the Antiderivative and The Fundamental Theorems of Calculus
- Summation
- Using Rectangles to Estimate the Area
- Using Infinitely Many Rectangles to Estimate the Area (Limit Definition of an Integral)
- The Antiderivative as a Function
- The Fundamental Theorem of Calculus
Chapter 6: Integration Techniques – Methods for Evaluating an Integral
- Basic Integration Formulas
- Numerical Integration and Estimating
- Substitution
- Integration by Parts
- Inverse Function Derivatives
- Trigonometric Derivatives
- Trigonometric Substitution
- Partial Fraction Decomposition
- Improper Integrals
- Integrals of Hyperbolic Functions
Chapter 7: Geometric Applications of the Integral – Using Integrals to Calculate Lengths, Areas, and Volumes
- Area between Curves
- Volumes of Revolution – Disk Method
- Volumes of Revolution – Cylindrical Shell Method
- Arc Length
- Surface Area
Chapter 8: Other Applications – Some Applications to the Sciences
- Work
- Moments and Centers of Mass
- Mass and Population Density
- Flow Rate and Poiseuille’s Law
- Present and Future Value
- Distrubution Functions
- Probability and Mean
Chapter 9: Sequences and Series – How to Identify Convergence of, and the uses for a Series
- Sequences
- Introduction to Series
- Integral and Comparison Tests
- Absolute and Conditional Convergence
- Ratio and Root Tests
- Power Series
- Taylor and Maclaurin Series
- Fourier Series
Chapter 10: Introduction to Differential Equations – Evaluating Basic First and Second Order Differential Equations
- Exponential Growth and Decay
- Slope Fields
- Classes of Differential Equations
- First Order Differential Equations
- Logistic Differential Equations
- Systems of Differential Equations
- Second Order Differential Equations