Calculus                            Download PDF

Credit Hours: 1.0

Course Length: 2 Semesters

Course Description

The Calculus course is a comprehensive look at the study of differential and integral calculus concepts including limits, derivative and integral computation, linearization, Riemann sums, the Fundamental Theorem of Calculus, and differential equations. Applications include graph analysis, linear motion, average value, area, volume, and growth and decay models.

Course Objectives

  • Evaluate limits, including those involving infinity.
  • Define and apply numerical and function derivatives.
  • Understand the relationship between continuity and differentiability.
  • Differentiate a variety of functions written in explicit and implicit form.
  • Analyze the behavior of a function using limits and derivatives.
  • Apply differential calculus to linear approximations, motion along a line, related rate problems, and optimization.
  • Define and apply antiderivatives and indefinite integrals.
  • Define and apply Riemann sums and definite integrals.
  • Understand the consequences of the Fundamental Theorem of Calculus.
  • Integrate a variety of functions.
  • Apply integral calculus to average value, total change, motion along a line, initial value problems, area, and volume.
  • Model and solve problems involving differential equations.

Course Prerequisites

Pre-Calculus

Required Textbook(s) and/or Materials

Calculator

Java is needed for the embedded graphing calculator applet (GCalc). A free download is available at http://www.java.com/en/download/

Course Outline

Module I: Limits and Continuity

  • Section A – Concept of a Limit
  • Section B – Algebraic Computation of a Limit
  • Section C – Limits Involving Infinity
  • Section D – Continuity
  • Section E – Intermediate Value Theorem

Module II: Derivatives

  • Section A – Concept of a Derivative
  • Section B – Differentiability
  • Section C – Graphs of fand f ′
  • Section D – Motion along a Line
  • Section E – Tangent Line Approximation

Module III: Differentiation

  • Section A – Basic Computation Rules
  • Section B – Higher Order Derivatives
  • Section C – Product, Quotient, and Chain Rules
  • Section D – Implicit Differentiation
  • Section E – Derivatives of Inverse Functions

Module IV: Graph Behavior

  • Section A – Asymptotes and End-Behavior
  • Section B – Increasing/Decreasing Behavior and Concavity
  • Section C – Relative Extreme Values and Points of Inflection
  • Section D – Absolute Extreme Values and Extreme Value Theorem
  • Section E – Graph Analysis

Module V: Derivative Applications

  • Section A – Mean Value and Rolle’s Theorems
  • Section B – Rates of Change
  • Section C – Related Rates
  • Section D – Optimization

Semester 1 Exam

Module VI: Antidifferentiation

  • Section A – Antiderivatives and Indefinite Integrals
  • Section B – Slope Fields
  • Section C – Basic Computation Rules
  • Section D – Substitution Rule
  • Section E – Initial Value Problems

Module VII: The Definite Integral

  • Section A – Area and the Riemann Sums
  • Section B – Approximation Methods
  • Section C – Fundamental Theorem of Calculus, Part 1
  • Section D – Computation of Definite Integrals
  • Section E – Fundamental Theorem of Calculus, Part 2

Module VIII: Integral Applications

  • Section A – Total Change
  • Section B – Average Value of a Function
  • Section C – Motion along a Line Revisited

Module IX: Area and Volume

  • Section A – Area between Two Curves
  • Section B – Volume of Solids Using Cross Sections
  • Section C – Volume of Solids of Revolution

Module X: Differential Equations and Their Applications

  • Section A- Separable Differential Equations
  • Section B- Modeling Using Differential Equations
  • Section C- Growth and Decay Models

Semester 2 Exam