# :: High School Courses

## Calculus A

### Course Description

Calculus A, the first of a two-semester course, centers on limits, differentiation, and applications of differentiation. Topics in this course apply to many problems studied in physics and engineering. Students review algebra concepts and learn fundamental calculus concepts along with working problems for limits and derivatives. Students apply rules for finding different derivatives as well as learn the applications of the derivative. After finding the area under a curve using several different methods, students will complete an essay assignment that applies this to a real-world problem. Students conclude the course by applying theorems and demonstrating knowledge of basic rules for antiderivatives.

### Course Organization

Modules. The Modules are practice assignments. Each Module is a non-graded exercise that allows you to build your knowledge and identify your strengths and weaknesses.

### Course Objectives

Upon completing of this course, you will be able to:

• identify and apply properties of sets;
• determine instantaneous rates of change of functions;
• apply the definition of the derivative;
• determine the symmetric difference and the difference quotient;
• determine the limits of functions;
• find the derivative of functions using various methodologies;
• determine the area under a curve;
• identify the concavity of a curve;
• identify critical points of curves;
• apply maximum and minimum to solve application problems;
• apply rules for finding antiderivatives;
• demonstrate knowledge of the Fundamental Theorem of Calculus; and
• apply the Second Fundamental Theorem to simplify definite integrals.

### Final Examination

The final examination is comprehensive; it covers the material from all of the units. To pass the course, you must receive a grade of 70 percent or better. You can apply to take the Final Exam after 100 percent of your graded assignments have been submitted, and at least 70 percent have been graded and returned to you.
Format: Objective
Time Allowed: 3 hours

### Course Outline

Total Number of Units: 6
Total Number of Activities: 50
Total Number of Graded Assignments: 20

Unit 1: Review and Fundamental Concepts
Module 1: Sets – Description, Roster, Rule
Module 2: Sets – Elements
Module 3: Sets – Intersection, Union, Power Sets
Module 4: Sets – Cross Product Set, Relation, Function
Module 5: Functions – Notation, Domain, and Range
Module 6: Functions – Linear Functions
Module 7: Functions – Quadratic Functions
Module 8: Functions – Polynomials
Module 9: Functions – Rational Functions
Module 10: Functions – Transcendental Functions
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Unit 1 Test – Review and Fundamental Concepts

Unit 2: Limits and the Derivative
Module 1: Slope and Rate of Change
Module 2: Difference Quotient and Symmetric Difference
Module 3: Intuitive Limits
Module 4: Formal Definitions of Limits
Module 5: Limit Theorems
Module 6: Instantaneous Rate of Change
Module 7: Definition of the Derivative
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Unit 2 Test – Limits and the Derivative

Unit 3: Rules for the Derivative
Module 1: Derivative of a Constant
Module 2: Derivative of a Positive Integer Power Rule
Module 3: Derivative of a Constant Times a Function
Module 4: Derivative of a Sum
Module 5: Derivative of a Difference
Module 6: Derivative of a Product
Module 7: Derivative of a Quotient
Module 8: Chain Rule
Module 9: Alternate Notation
Module 10: Power Rule
Module 11: Table of Rules
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Table of Rules

Calculus A Midterm Exam
This Midterm Exam counts as a graded assignment and does not need to be taken at a testing center.

Unit 4: Applications of the Derivative
Module 1: Maxima and Minima
Module 2: Second and Higher Order Derivatives
Module 3: Concavity and Inflection Points
Module 4: Critical Points
Module 5: Maxima and Minima Problems
Module 6: Implicit Differentiation
Module 7: Related Rates
Module 8: Linear Approximation
Module 9: Newton’s Method
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Unit 4 Test – Applications of the Derivative

Unit 5: The Area Problem
Module 1: Summation Formulas and Mathematical Induction
Module 2: Estimating Area on a Grid
Module 3: Estimating the Area under a Curve by Summing Rectangles
Module 4: Estimating the Area under a Curve by Summing Trapezoids
Module 5: Estimating the Area under a Curve by Simpson’s Rule
Module 6: Limit of a Summation
Module 7: The Definite Integra
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Area Essay Assignment
Graded Assignment 4: Unit 5 Test – The Area Problem

Unit 6: Methods of Integration
Module 1: The Antiderivative
Module 2: Basic Rules for Antiderivatives
Module 3: Substitution
Module 4: Fundamentals Theorem of Calculus
Module 5: Algebraic and Geometric Areas
Module 6: Second Fundamental Theorem of Calculus
Graded Assignment 1: Homework Assignment #1
Graded Assignment 2: Homework Assignment #2
Graded Assignment 3: Unit 6 Test – Methods of Integration

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