# :: 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.

**Graded Assignments.** Graded Assignments are the tasks you will submit to your instructor for a grade. Each assignment contains specific information about how your work will be assessed and how credit will be given for your responses. The average of your assignments will count for 75 percent of your course grade.

### 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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