Course Timeline for College Calculus

Chapter P:Preparation for Calculus

P-1 Graphs and Models

  • Viewing and interpreting graphs
  • Points of intersection
  • Symmetry

P-3 Functions and Their Graphs

  • Functions
  • Piece-wise functions
  • Even and odd functions
  • Composition of functions

P-4 Fitting Models to Data

  • Linear model with real-life data
  • Quadratic model with real-life data
  • Trigonometric model with real-life data

Chapter 1: Limits and Their Properties

1.1A Preview of Calculus

1.2Finding Limits Graphically and Numerically

  • Definition
  • Properties
  • Evaluating Limits Analytically
  • Trig Limits
  • Limits with radicals
  • Limits of composition functions
  • Limits of functions that agree at all but one point
  • Continuity and One-Sided Limits
  • One and two sided limits
  • Removable discontinuity
  • Nonremovable discontinuity – Jump, asymptote, or oscillating
  • Intermediate Value Theorem for continuous functions
  • Infinite Limits
  • Asymptotic behavior
  • End behavior
  • Visualizing limits

Chapter 2: Differentiation

2.1 The Derivative and the Tangent Line Problem

  • Tangent to a curve
  • Slope of a curve
  • Normal to a curve
  • Definition

2.2 Basic Differentiation Rules and Rates of Change

  • Constant, Power, Sum and Difference, and Constant Multiple Rules
  • Sine and Cosine
  • Rates of Change

2.3 Product and Quotient Rules and Higher-Order Derivatives

  • Product and Quotient Rules
  • Trigonometric functions
  • Second and higher order derivatives
  • Acceleration due to gravity

2.4 The Chain Rule

  • Composition of a function
  • Power Rule
  • Trig functions with the Chain Rule

2.5 Implicit Differentiation

  • Implicit and Explicit functions
  • Differential method
  • Second derivative implicitly
  • Slope, tangent, and normal

2.6 Related Rates

  • Applications to derivatives
  • Guidelines for related rate problems

Chapter 3: Applications of Differentiation

3.1 Extrema on an Interval

  • Relative extrema
  • Critical numbers
  • Finding extrema on a closed interval
  • Absolute extrema

3.2 Rolle’s Theorem and the Mean Value Theorem

  • Illustrating Rolle’s Theorem
  • Tangent line problems and instantaneous rate of change problems with the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

  • Testing for increasing and decreasing
  • First Derivative Test for extrema
  • Applications

3.4 Concavity and the Second Derivative Test

  • Testing for concavity
  • Points of inflection
  • Second Derivative Test for extrema

3.5 Limits at Infinity

  • Horizontal Asymptotes
  • Limits at infinity
  • Trig functions
  • Infinite limits at infinity

3.6 A summary of Curve Sketching

  • Rational functions
  • Radical functions
  • Polynomial function
  • Trig function

3.8 Newton’s Method

  • Approximate zeros

3.9 Differentials

  • Tangent line approximation
  • Error propagation

Chapter 4: Integration

4.1 Antiderivatives and Indefinite Integration

  • Definition
  • Integration Rules
  • Vertical Motion

4.2 Area

  • Sigma notation
  • Upper and lower sums

4.3 Riemann Sums and Definite Integrals

  • Subintervals with equal and unequal widths
  • Definition
  • Continuity
  • Area of a region
  • Properties of definite integrals

4.4 The Fundamental Theorem of Calculus

  • Guidelines for using FTC
  • Mean Value Theorem for integrals
  • Average Value of a function
  • Second fundamental theorem

4.5 Integration by Substitution

  • Composition function
  • Change of variables
  • Power rule for integration

4.6 Numerical Integration

  • Trapezoidal Rule
  • Simpson’s Rule

Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions, Chapter 6: Differential Equations

5.1 The Natural Logarithmic Function: Differentiation

  • Definition
  • Properties of the Natural Logarithmic Function
  • Definition of e
  • Derivative of ln

5.2 The Natural Logarithmic Function: Integration

  • Log rule for integration
  • Trig functions

5.3 Inverse Function

5.4 Exponential Functions: Differentiation and Integration

  • Definition of
  • Operations and properties with exponential functions

6.1 Slope Fields and Euler’s Method

  • General and particular solutions
  • Slope fields – Visualizing and sketching
  • Approximating solutions with Euler’s method

6.2 Differential Equations: Growth and Decay

  • Growth and decay model

Chapter 7: Applications of Integration

7.1 Area of a Region Between Two Curves

  • Area between two curves
  • Intersecting curves

7.2 Volume: The Disk Method

  • Disk and washer method

7.3 Volume: The Shell Method