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Pre-Calculus

• Linear Equations
• Solving Polynomial Equations
• Functions and Domain
• Piecewise Functions
• Topics in Trigonometry
• Exponentials and Logarithms

Limits and Continuity

• The Three Tiers of Limits
  • Tier One: Plug It in
  • Tier Two: Algebra Gymnastics
  • Tier Three: Vertical Asymptote
• Properties of Limits
• Continuity
• Intermediate Value Theorem
• Limits to Infinity
• Analyzing Functions using Limits

Derivatives

• Definition of the Derivative
• Tanget Lines / Normal Lines
• Differentiability
• Power Rule
• Different Derivatives
• Product Rule
• Quotient Rule
• Chain Rule
• Implicit Differentiation
• Derivatives of Inverse Functions
• Derivatives and Physics

Applications of Derivatives

• Rates of Change
• Increasing / Decreasing
• Maxima and Minima
• First Derivative Analysis
• Concavity and Inflection Points
• First and Second Derivative Analysis
• Graphical Interpretations
• Mean Value Theorem
• Optimization
• Related Rates
• Linearization
• L'Hospital's Rule

Integrals

• Estimating the Integral Using
  • LRAM/ RRAM / MRAM
  • Trapezoidal Rule
• Definition of the Integral
• Integration Using Geometry
• Anti-Derivatives
• Definite Integrals
• Average Value of a Function
• Fundamental Theorem of Calculus
• Integration by Substitution
• Slope Fields
• Initial Value Problems
• Differential Equations

Applications of Integrals

• Exponential Growth and Decay
• Net Change
• Integrals and Physics
• Areas Between Curves
• Volumes of Solids Given a Base and Cross Sections
• Solids by Revolution:
  • Disc Method
  • Washer Method

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Beyond Calculus is designed for students currently taking Calculus.

About the Author

Integrals as Net Change

 

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The Big Picture

 

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Net Change and Antiderivatives

 

Images taken from the AP©

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Net Change and the Graph of f '

 

Images taken from the AP©

 

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Practice Problems