# Calculus I/II

## Course Overview

Calculus I/II is a comprehensive Calculus course designed to cover Limits, Derivatives, Integrals, and Sequences and Series. Because the sheer volume of material is so great, the course is designed to take an entire school year and is intended for students with a proven record of accomplishment and interest in mathematics. The course is also designed to include 1-2 weeks at the end of the class for preparation for the AP Calculus BC Exam, at the instructor’s prerogative. Specific topics covered at the beginning of the course are included below. After differential calculus, students then advance to Integral Calculus. Integral Calculus emphasizes techniques of integration and its applications. Once students have shown competence with both integral and differential Calculus, they are prepared for the final two concepts of the course: Differential Equations and Sequences and Series. The Differential Equations section is designed to provide students with an introduction to the topic, which will give them an advantage when they take a more formal Differential Equations course. Sequences and Series mainly focuses on Tests for Convergence and Divergence, Power Series, Writing Functions as a Power Series, and Taylor/Maclaurin Series. This concludes the content for the course.

## Course Content

### Unit 1: A Review of the Basic Functions, Limits and Continuity, Graphical Derivatives and an Introduction to Derivatives

As limits and continuity are the foundation for all of Differential Calculus, this is perhaps the most important unit in which students need to be proficient. The main emphasis of the unit is to prepare students to take derivatives and to consider graphs, limits, continuity and derivatives from a graphical, numerical and algebraic perspective. The following topics are covered in Unit 1:

• Review of functions from advanced Algebra and Trigonometry
• Review of Laws of Logarithms and Exponents
• An exploration of the line tangent to a function linking velocity and tangent lines
• An exploration of limits using epsilons and deltas
• Evaluation of limits using graphs, tables and graphing calculators
• Evaluation of limits using limit laws
• Evaluation of one-sided limits
• Using limits to determine continuity of a function
• Evaluate limits at infinity and link to horizontal and vertical asymptotes
• Determine derivatives using the Limit Definitions of Derivative
• Determine equations of lines tangent to functions using Limit Definitions of Derivatives
• Explore higher derivatives
• Relate graphs of functions and their derivatives based on behaviors of the functions
• Explore graphs of higher order derivatives

### Unit 2: Rules for Differentiation

This unit provides students with the skills to take the derivative of all continuous functions that they have encountered thus far in their mathematical studies. It is the most important unit for differential calculus insofar as they can’t successfully move onto subsequent topics if they can’t take a derivative properly. The following topics are covered in Unit 2:

• Derivatives of polynomials and exponential functions
• Product and quotient rules
• Derivatives of trigonometric functions
• Chain rule Implicit differentiation
• Derivatives of logarithmic functions
• Related rates of change
• Applications of rates of change in science

### Unit 3: Applications of Derivatives

Students learn to take derivatives so they can apply them to solve real-world and theoretical problems. This unit involves using derivatives to better understand graph theory, optimization, L’Hospitals Rule and Newton’s Method and represents the acme of differential calculus in two dimensional space. The unit concludes with an exploration of antiderivatives and an introduction to integrals. This unit covers the following topics:

• Linearization and differentials
• Maximum and minimum values of a function
• The Mean Value Theorem
• Derivatives and the shapes of graphs
• Indeterminate forms and L’Hospital’s Rule
• Summary of curve sketching using derivatives
• Graphing with calculus and calculators
• Optimization Newton’s Method Antiderivatives

### Unit 4: Integrals

Much like Differential Calculus, Integral Calculus has far-reaching applications in the sciences. This unit provides students with both the theoretical and mechanical/algebraic skills to fully understand how and why integrals are taken. This unit covers the following topics:

• Areas and distances
• The definite integral
• The Fundamental Theorem of Calculus
• Indefinite integrals and the Net Change Theorem
• Integration by substitution

### Unit 5: Applications of Integrals

Much like differential calculus, students learn integral calculus so they can use techniques to solve real-world problems. This unit does not use advanced techniques of integration, which are learned in Unit 6. This unit covers the following topics:

• Areas between curves
• Volumes of solids of revolution using the disk/washer method
• Volumes of solids of known cross section
• Volumes of solids of revolution using the shell method
• Work
• Average value of a function

### Unit 6: Techniques of Integration

This unit concentrates on more advanced techniques of integration. This unit covers the following topics:

• Integration by parts
• Trigonometric Integrals
• Trigonometric substitution
• Partial fraction decomposition
• Strategies of integration
• Integration using tables and calculators
• Numerical approximation of integration
• Improper integrals

### Unit 7: Calculus of Polar and Parametric Equations and Differential Equations

This unit rounds out calculus of two-dimensional spaces, introducing students to using calculus on less traditional functions such as parametric and polar functions. The unit also includes and introduction to differential equations. The following topics are included in the unit:

• Parametric equations
• Arc Lengths of Curves
• Calculus of parametric equations and curves
• Polar coordinates and graphs
• Calculus of polar equations
• The definition of a differential equation
• Direction/slope fields and Euler’s Method
• Exponential growth and decay
• Separable differential equations
• Models for population growth

### Unit 8: Sequences and Series

This is the final unit in the course and can be the most challenging for students, as it if probably the most theoretical. The unit explores all aspects of convergence and divergence of sequences and series, and concludes with writing functions as power series using Taylor and Maclaurin polynomials. The unit covers the following topics:

• Sequences
• Series
• The integral test
• The comparison and limit comparison tests
• Alternating series
• The ratio and root test
• Practice determining convergence and divergence of series
• Power series
• Representing functions as power series
• Taylor and Maclaurin series
• Lagrange error bounds