Course: Calculus

Table of Contents

• Defining Limits
  • Limits and Limit Notation
  • Limits from Tables and Graphs
  • Limits by Direct Substitution
  • Properties of Limits
• Evaluating Limits
  • Evaluating Limits Using Algebraic Techniques
  • One-Sided Limits
  • Existence of Limits
  • The Squeeze Theorem
  • Limits of Trigonometric Functions
  • Limits at Infinity
  • Horizontal and Vertical Asymptotes of a Function
  • Limits and Asymptotic Behavior
• Continuity
  • Continuity at a Point
  • Classifying Discontinuities
  • Continuity of Functions
  • Intermediate Value Theorem
• Defining Derivatives
  • Average and Instantaneous Rates of Change
  • Definition of the Derivative
  • The Differentiability of a Function
  • Estimating Derivatives
• Differentiation Rules
  • Power Rule of Derivatives
  • The Product Rule
  • The Quotient Rule
  • Differentiation of Trigonometric Functions
  • The Chain Rule
  • Differentiation of Reciprocal Trigonometric Functions
  • Second- and Higher-Order Derivatives
  • Implicit Differentiation
  • Differentiation of Exponential Functions
  • Differentiation of Inverse Functions
  • Differentiation of Logarithmic Functions
  • Derivatives of Inverse Trigonometric Functions
  • Combining the Product, Quotient, and Chain Rules
• Applications of Differentiation
  • Rate of Change and Derivatives
  • Related Time Rates
  • Equations of Tangent Lines and Normal Lines
  • Linear Approximation
  • The Mean Value Theorem
  • Increasing and Decreasing Intervals of a Function Using Derivatives
  • Critical Points and Local Extrema of a Function
  • Convexity and Points of Inflection
  • Second Derivative Test for Local Extrema
  • Absolute Extrema
  • Interpreting Graphs of Derivatives
  • Graphing Using Derivatives
  • Optimization: Applications on Extreme Values
  • Applications of Derivatives on Rectilinear Motion
  • L’Hôpital’s Rule
  • Comparing Rate of Growth of Functions
• Indefinite Integrals
  • Antiderivatives
  • Indefinite Integrals: The Power Rule
  • Indefinite Integrals: Trigonometric Functions
  • Indefinite Integrals: Exponential and Reciprocal Functions
  • Indefinite Integrals and Initial Value Problems
• Definite Integrals
  • Riemann Sums
  • Riemann Sums and Sigma Notation
  • Definite Integrals as Limits of Riemann Sums
  • Numerical Integration: Riemann Sums
  • Numerical Integration: The Trapezoidal Rule
  • Properties of Definite Integrals
• Fundamental Theorem of Calculus
  • The Fundamental Theorem of Calculus: Functions Defined by Integrals
  • The Fundamental Theorem of Calculus: Evaluating Definite Integrals
  • The Net Change Theorem
• Integration Skills
  • Integration by Substitution: Indefinite Integrals
  • Integration by Substitution: Definite Integrals
  • Integrals Resulting in Logarithmic Functions
  • Integrals Resulting in Inverse Trigonometric Functions
  • Integration by Parts
  • Integration by Partial Fractions with Linear Factors
  • Improper Integrals: Infinite Limits of Integration
  • Improper Integrals: Discontinuous Integrands
• Applications of Integration
  • Average Value of a Function
  • Area between a Curve and a Line
  • Area between Curves
  • Rectilinear Motion and Integration
  • Volumes of Solids of Revolution
  • Volumes by Slicing
  • Arc Length by Integration
• Differential Equations
  • Basics of Differential Equations
  • Slope Fields and Solution Curves
  • Euler’s Method
  • Separable Differential Equations
  • Initial Value Problems
  • Exponential Growth and Decay Models
  • The Logistic Model
• Parametric Equations
  • Parametric Equations and Curves in Two Dimensions
  • Conversion between Parametric and Rectangular Equations
  • Derivatives of Parametric Equations
  • Second Derivatives of Parametric Equations
  • Arc Length of Parametric Curves
  • Planar Motion Using Parametric Equations
• Polar Coordinates
  • Polar Coordinates
  • Conversion between Rectangular and Polar Equations
  • Slope of a Polar Curve
  • Area Bounded by Polar Curves
• Vector-Valued Functions
  • Vector-Valued Functions
  • Derivatives of Vector-Valued Functions
  • Integrals of Vector-Valued Functions
• Infinite Series
  • Infinite Series
  • Partial Sums
  • Infinite Geometric Series
  • The nth Term Divergence Test
  • Integral Test for Series
  • Harmonic and p-Series
  • Comparison Test for Series
  • Limit Comparison Test
  • Alternating Series Test
  • Conditional and Absolute Convergence
  • Ratio Test
  • Remainder of an Alternating Series
• Power Series
  • Power Series and Radius of Convergence
  • Operations on Power Series
  • Representing Rational Functions Using Power Series
  • Differentiating and Integrating Power Series
• Taylor and Maclaurin Series
  • Taylor Polynomials Approximation to a Function
  • Taylor Series
  • Lagrange Error Bound
  • Maclaurin Series
  • Maclaurin and Taylor Series of Common Functions