Calculus 1 Notes

Chapter 1- Various Important Functions

• Review of Earlier Functions
  • Linear Functions (Lines)
  • Quadratic Functions (Parabolas)
  • General Polynomial Functions
  • Exponential Functions
  • Logarithmic Functions
  • Trigonometric Functions
• Inverse Functions
• Piecewise Functions
• Hyperbolic Trigonometric Functions

Chapter 2- Limits

• Introduction to Limits
• Right and Left Hand Limits
• Definition of the Limit
• Graphical Limits
• Tabular Limits
• Direct Substitution
• Properties of Limits
• Indeterminate Forms
• Analytical Procedures for Computing Limits
  • Factoring and Cancellation
  • Addition of Fractions
  • Conjugate Multiplication
• Infinite Limits
• Epsilon-Delta Definition of the Limit
• Squeeze-Theorem

Chapter 3- Continuity

• Introduction to Continuity
• Definition of Continuity
• Continuous Functions
• Discontinuities in Graphs
• Discontinuities in Functions

Chapter 4- The Derivative: Introduction and Basic Methods

• Average Rate of Change
• Instantaneous Rates of Change and the Tangent Line
• The Definition of the Derivative
• Differentiability
• Examples Using the Derivative Definition
• The Power Rule
• Properties of the Derivative

• The Product Rule

• The Quotient Rule

• The Chain Rule

Chapter 5- The Derivative: Advanced Methods

• Trig Function Derivatives
• Inverse Trig Derivatives
• Hyperbolic Trig Derivatives
• Exponential Derivatives
• Logarithmic Derivatives
• Implicit Differentiation
• Logarithmic Differentiation Method
• Partial Derivatives

Chapter 6- Applications of Derivatives

• Intermediate Value Theorem
• Mean Value Theorem
  • Rolle’s Theorem
• Introduction to Extrema
• Local Extrema
• The First Derivative Test
• Absolute Extrema
• Optimization
• Higher order Derivatives
• Second Derivative Test
• Concavity and Inflection Points
• Rectilinear Motion: Position, Velocity and Acceleration
• Introduction to Differential Equations
  • A Few Basic Examples
  • Local Linearity and Euler’s Method
• L’Hôpital’s rule
• Related Rates

Chapter 7- The Integral: Introduction and Basic Methods

• Introduction to the Area Under Curves
• Area Approximation techniques: Riemann Sums
  • Left and Right Endpoint Rectangles
  • Center-Aligned Rectangles
  • The Trapezoid Rule
  • Simpson’s Rule
  • The Exact Area Using Riemann Sums
• The Antiderivative
• The Indefinite Integral
• The Integration Power Rule
• Properties of Integrals
• The Definite Integral
• The Fundamental Theorem of Calculus
• Integral Definition Summary (With Examples)

Chapter 8- Advanced Integration Techniques

• Advanced Power Rule Problems
• Reverse Chain Rule
• U-Substitution
• Long Division
• Completing the Square
• More Examples

Chapter 9- Integration Applications

• Average Values
• Arc Length
• Area Between Curves (Over x)
• Area Between Curves (Over y)
• Volumes of Revolution
  • Disk Method
  • Washer Method
  • Cylindrical Shell Method
  • Multiple Integral Examples
• Volumes of Known Cross Section
• Rectilinear Motion Revisited