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- Preview of Calculus
o Definition of Calculus
o Definition of Differential Calculus
o Definition of Integral Calculus
- Finding Limits Graphically and Numerically
o Finding Limits Numerically
o Finding Limits Graphically
o The Definition of Limits
- Evaluating Limits Analytically
o Theorems of Limits.
- Continuity and One-Sided Limits
o Definition of Continuity
o Definition of Discontinuity
o Types of Discontinuity
- Removable Discontinuity
- Non-Removable Discontinuity
o One-Sided Limits
o Continuity on a Closed Interval
o Intermediate Value Theorem
- Infinite Limits
o Definition of Infinite Limits
o Vertical Asymptote
o Properties of Infinite Limits
- The Derivative and The Tangent Line Problem
o The Difference between a Secant Line and Tangent Line
o First Derivative of a Function
o Finding The Slope and The Equation of a Tangent Line
o Alternative Form of a Derivative
- Differentiation Rules and Rates of Change
o Theorems of Derivatives
- Constant Rule
- Power Rule
- Constant Multiplication Rule
o Derivatives of Trigonometry
- Derivatives of Sine and Cosine and their Proofs
o Derivatives of Natural Logarithms
o Horizontal Tangent
o Position Function
- Rates of Change
- Average Velocity
- Instantaneous Velocity
- Speed
- Acceleration
- Free Falling Objects
- Product and Quotient Rules and Higher Order Derivatives
o Power Rule
o Quotient Rule
o Derivatives of Trigonometry
- Derivatives of Secant and Cosecant and their Proofs
- Derivatives of Tangent and Cotangent and their Proofs
o Higher Order Derivatives
- Second Derivative
- Third Derivative and etc.
o Business Applications
o Cost Function and Average Cost Function
o Revenue Function and Average Revenue Function
o Profit Function and Average Profit Function
- Chain Rule and Derivative of Logarithmic Function
o Chain Rule ( Derivative of Composite Function )
o Derivative of Natural Exponential Function
o Derivatives of Natural Logarithmic Function
o Derivative of Natural Logarithmic Function of Absolute Value
o Derivative of Exponential Function
o Derivatives of Common Logarithmic Function
- Implicit Differentiation
o Difference of Explicit and Implicit Differentiation
o Implicit Differentiation
o Implicit Logarithmic Differentiation
- Derivatives of Inverse Function
o Derivatives of Inverse Functions
o Derivatives of Inverse Trigonometric Functions and their Proofs
- Related Rates and Application Problems
o Definition of Related Rates
o Application Problems of Related Rates
- Newton’s Method
o Definition of Newton’s Method
- Extreme on an Interval
o The Extreme value Theorem
o Relative (Local) Maximum / Minimum
o Absolute (Global) Maximum / Minimum
o Extreme on an Interval
o Critical Values / Numbers
o Critical Points
- Rolle’s Theorem and Mean Value Theorem
o Definition of Rolle’s Theorem
o Definition of Mean Value Theorem
- Increasing and Decreasing Functions
and First Derivative Test.
o Definition of Increasing and Decreasing Functions
o Theorem for Increasing and Decreasing Functions
o Guidelines (Steps) for Finding Increasing and Decreasing Functions
o First Derivative Test
- Concavity and Second Derivative Test
o Theorem of Second Derivative
o Second Derivative Test
o Concavity ( Concave upward and Concave downward )
o Point of Inflection
- Limits at Infinity and Horizontal Asymptote
o Definition of Limits of Infinity
o Theorems of Limits of Infinity
o Horizontal Asymptotes
o Limits of Infinity for Square Root Functions
o Limits of Infinity for Trigonometric Functions
o Limits of Infinity for Logarithmic Functions
- Graph and Curve Sketching
o Guideline for analyzing the Graph and Curve Sketching of a Function
- Optimization Problems
o Applications of Differential Calculus
o Guideline for Solving Optimization Problems
- Optimization Problems for Business and Economics
o Revenue maximization
o Cost Minimization
o Profit Maximization
o Setting for Price selling
- Differentials and Linear Approximation
o Linear Approximation (Tangent line Approximation)
o Differentials (Differential Approximation)
o Differential Approximation error ( Estimation of Error )
o Propagated Error
o Relative Error
o Percentage Relative Error ( Percent Error )
o Calculating Differential Approximation without Using Calculator