Modular Calculus - Math 1411

Course Content

The texbook is: Calculus, 8th edition, by Larson, Hostetler.

Module 1

Chapters 1 and 2
Section 1.1 - A Preview of Calculus
  • What is Calculus?
  • The Tangent Line Problem
  • The Area Problem
Section 1.2 - Finding Limits Graphically and Numerically
  • An Introduction to Limits
  • Estimating a Limit Numerically
  • Finding a Limit
  • Limits That Fail to Exist
  • Behavior That Differs From the Right and Left
  • Unbounded Behavior
  • Oscillating Behavior
  • A Formal Definition of Limit
  • Finding a δ for a given ε
  • Using the Definition of Limit
  • Common Pitfalls in Finding Limits
Section 1.3 - Evaluating Limits Analytically
  • Properties of Limits
  • Some Basic Limits
  • Evaluating Basic Limits
  • The Limit of a Polynomial
  • The Limit of a Rational Function
  • The Limit of a Composite Function
  • Limits of Trigonometric Functions
  • A Strategy for Finding Limits
  • Functions That Agree at All But One Point
  • Rationalization Technique
  • The Squeeze Theorem
  • Two Special Trigonometric Limits
  • Technology Pitfalls
Section 1.4 - Continuity and One-Side Limits
  • Continuity at a Point and on an Open Interval
  • Definition of Continuity
  • Continuity of a Function
  • One-Sided Limits and Continuity on a Closed Interval
  • A One-Sided Limit
  • The Greatest Integer Function
  • The Existence of a Limit
  • Definition of Continuity on a Closed Interval
  • Continuity on a Closed Interval
  • Charles Law and Absolute Zero
  • Properties of Continuity
  • Applying Properties of Continuity
  • Continuity of a Composite Function
  • Testing for Continuity
  • The Intermediate Value Theorem
  • Application of the Intermediate Value Theorem
Section 1.5 - Infinite Limits
  • Infinite Limits
  • Definition of Infinite Limits
  • Determining Infinite Limits from a Graph
  • Vertical Asymptotes
  • Properties of Infinite Limits
  • Determining Infinite Limits
Section 2.1 - The Derivative and the Tangent Line Problem
  • The Tangent Line Problem
  • Definition of Tangent Line with Slope m
  • Relationship Between the Secant Line and the Tangent Line
  • The Slope of the Graph of a Linear Function
  • Tangent Lines to the Graphs of Nonlinear Functions
  • Definition of the Derivative of a Function
  • Differentiable Function
  • Finding the Derivative by the Limit Process
  • Using the Derivative to Find the Slope at a Point
  • Differentiability and Continuity
  • A Graph With a Sharp Turn
  • A graph With a Vertical Tangent Line
  • Differentiability Implies Continuity
Section 2.2 - Basic Differentiation Rules and Rates of Change
  • The Constant Rule
  • Using the Constant Rule
  • The Power Rule
  • Using the Power Rule
  • Finding the Slope of a Graph
  • Finding the Equation of a Tangent Line
  • The Constant Multiple Rule
  • Using the Constant Multiple Rule
  • The Sum and Difference Rules
  • Using the Sum and Difference Rules
  • Derivatives of Sine and Cosine Functions
  • Rates of Change
  • Finding Average Velocity of a Falling Objectr
  • Using the Derivative to Find Velocity
Section 2.3 - Product and Quotient Rules and Higher-Order Derivatives
  • The Product Rule
  • Proving the Product Rule
  • Using the Product Rule
  • The Quotient Rule
  • Proving the Quotient Rule
  • Using the Quotient Rule
  • Proving the Power Rule
  • Derivatives of Trigonometric Functions
  • Higher-Order Derivatives
  • Finding the Acceleration Due to Gravity
Section 2.4 - The Chain Rule
  • Find the Derivative of a Composite Function Using the Chain Rule
  • Find the Derivative of a function using the General Power Rule
  • Simplify the Derivative of a Function Using Algebra
  • Find the Derivative of a Trigonometric Function Using the Chain Rule
  • The Chain Rule
  • Proving the Chain Rule
  • Decomposition of a Composite Function
  • The General Power Rule
  • Applying The General Power Rule
  • Differentiating Functions Involving Radical
  • Differentiating Functions with Constant Numerators.
  • Simplifying Derivatives by Factor Out the Least Powers
  • Simplifying the Derivative of a Quotient
  • Simplifying the Derivative of a Power
  • Repeated Application of the Chain Rule
  • Trigonometric Functions and the Chain Rule
  • Applying the Chain Rule to Trigonometric Functions
  • Tangent Line of a Trigonometric Function
Section 2.5 - Implicit Differentiation
  • Distinguish Between Functions Written in Implicit and Explicity Form
  • Use Implicit Differentiation to Find the Derivatives of a Function
  • Implicit and Explicit Functions
  • Finding the Slope of a Graph Implicitly
  • Finding the Second Derivative Implicitly
Section 2.6 - Related Rates
  • Find a Related Rate
  • Use Related Rates to Solve Real-Life Problems

Module 2

Chapters 3, Sections 4.1 and 4.2
Section 3.1 - Extrema on an Interval
  • Understand the Definition of Extrema of a Function on an Interval
  • Understand the Definition of Relative Extrema of a Function on an Open Interval
  • Find Extrema on a Closed Interval
  • Definition of Extrema
  • The Extreme Value Theorem
  • Relative Extrema and Critical Numbers
  • Definition of Relative Extrema
  • Definition of Critical Number
  • Relative Extrema Occur Only at Critical Numbers
  • Finding Extrema on a Closed Interval
Section 3.2 - Rolle's Theorem and the Mean Value Theorem
  • Understand and Use Rolle's Theorem
  • Understand and Use the Mean Value Theorem
  • Rolle's Theorem
  • Illustrating Rolle's Theorem
  • The Mean Value Theorem
  • Finding an Instantaneous Rate of Change
Section 3.3 - Increasing and Decreasing Functions and the First Derivative Test
  • Determine Intervals on Which a Function is Increasing or Decreasing
  • Apply the First Derivative Test to Find Relative Extrema on an Interval
  • Definition of Increasing and Deceasing Functions
  • Test for Increasing and Decreasing Functions
  • Guidelines for Finding Intervals on Which a Function is Increasing or Decreasing.
  • The First Derivative Test
  • Applying the First Derivative Test
Section 3.4 - Concavity and the Second Derivative Test
  • Determine Intervals on Which a Function is Concave Upward or concave Downward
  • Find any Points of Inflection of the Graph of a Function
  • Apply the Second Derivative Test to Find Relative Extrema of a Function
  • Definition of Concavity
  • Test for Concavity
  • Determining Concavity
  • Definition of Point of Inflection
  • Finding Points of Inflection
Section 3.5 - Limits at Infinity
  • Definition of Limits at Infinity
  • Definition of Horizontal Asymptote
  • Limits at Infinity
  • Finding a Limit at Infinity
  • Guidelines for Finding Limits at infinity of Rational Functions
  • Limits Involving Trigonometric Functions
  • Definition of Infinite Limits at Infinity
  • Finding Infinite Limits at Infinity
Section 3.6 - A Summary of Curve Sketching
  • Analyzing the Graph of a Function
  • Sketching the Graph of a Rational Function
  • Sketching the Graph of a Radical Function
  • Sketching the Graph of a Polynomial Function
  • Sketching the Graph of a Trigonometric Function
Section 3.7 - Optimization Problems
  • Solve Applied Minimum and Maximum Problems
  • Guidelines for Solving Applied Minimum and Maximum Problems
Section 3.8 - Newton's Method
  • Approximate a Zero of a Function Using Newton's Method
  • Using Newton's Method
  • An Example in Which Newton's Method Fails
  • Algebraic Solutions to Polynomial Equations
Section 3.9 - Differentials
  • Understand the Concept of Tangent Line Approximations
  • Using a Tangent Line Approximation
  • Definitions of Differentials
  • Error Propagation
  • Estimation of Error
  • Calculating Differentials
  • Finding the Differential of a Composite Function
  • Approximating Function Values
Section 4.1 - Antiderivatives and Indefinite Integration
  • Antiderivatives
  • Definition of Antiderivatuve
  • Solving Differential Equations
  • Notation for Antiderivatives
  • Basic Integration Rules
  • Applying Basic Integration Rules
  • Initial Conditions and Particular Solutions
  • Write the General Solution of a Differential Equation
  • Use Indefinite Integral Notation for Antiderivatives
  • Use Basic Integration Rules to find Antiderivatives
  • Find a Particular Solution of a Differential Equation
Section 4.2 - Area
  • Sigma Notation
  • Examples of Sigma Notation
  • Use Sigma Notation to Write and Evaluate a Sum
  • Summation Formulas
  • Understand the Concept of Area
  • Approximate The Area of a Plane Region
  • Upper and Lower Sums
  • Limits of the Lower and Upper Sums
  • Definition of the Area of a Region in the Plane
  • Finding Area of a Plane Using Limits

Module 3

Sections 4.3, 4.4, 4.5, 4.6, and Chapter 5 and 6
Section 4.3 - Riemann Sums and Definite Integrals
  • Definition of Riemann Sums
  • Understand the Definition of Riemann Sums
  • Definition of a Definite Integrals
  • Continuity Implies Integrability
  • Evaluating a Definite Integral as a Limit
  • The Definite Integral as the Area of a Region
  • Areas of Common Geometric Figures
  • Properties of Definite Integrals
  • Evaluating Definite Integrals
  • Additive Interval Property
  • Using the Additive Interval Property
  • Preservation of Inequaliaty
Section 4.4 - The Fundamental Theorem of Calculus
  • Discussion of the Fundamental Theorem of Calculus
  • Guidelines for Using the Fundamental Theorem of Calculus
  • Evaluate a Definite Integral Using the Fundamental Theorem of Calculus
  • Use the Fundamental Theorem of Calculus to Find Area
  • Understand and Use The Mean Value Theorem for Integrals
  • Find the Average Value of a Function Over a Closed Interval
  • Discussion of the Second Fundamental Theorem of Calculus
  • Understand and Use the Second Fundamental Theorem of Calculus
  • The Definite Integral as a Function
Section 4.5 - Integration by Substitution
  • Use Pattern Recognition to Find an Indefinite Integral
  • Antidifferentiation of a Composite Function
  • Change of Variables
  • Guidelines for Making a Change of Variables
  • The General Power Rule for Integration
  • Substitution and the General Power Rule
  • Use the General Power Rule for Integration to find an Indefinite Integral
  • Change of Variables for Definite Integrals
  • Use Change of Variables to Evaluate a Definite Integral
  • Use a Change of Variables to Find an Indefinite Integral
  • Integration of Even and Odd Functions
  • Evaluate a Definite Integral Involving Even and Odd Functiosn
Section 4.6 - Numerical Integration
  • The Trapezoidal Rule
  • Approximate a Definite Integral Using the Trapezoidal Rule
  • Simpson's Rule
  • Approximate a Definite Integral using Simpson's Rule
  • Approximations with Simpson's Rule
  • Analyze the Approximate Errors in the Trapezoidal Rule and Simpson's Rule
Section 5.1 - The Natural Logarithmic Function: Differentiation
  • Definition of the Natural Logarithmic Function
  • Properties of the Natural Logarithmic Function
  • Expanding Logarithmic Functions
  • Definition of the Number e
  • Derivative of the Natural Logarithmic Function
  • Logarithmic Differenetiation
  • Derivatives Using Absolute Value
  • Finding Relative Extrema
Section 5.2 - The Natural Logarithmic Function: Integration
  • Log Rule for Integration
  • Using the Log Rule for Integration
  • Using the Log Rule with a Change of Variables
  • Finding Area with the Log Rule
  • Guidelines for Integration
  • Substitution and the Log Rule
  • Trigonometric Identities
  • Using Trigonometric Identities
  • Derivative of the Secant Function
  • Integrating Trigonometric Functions
  • Find the Average Value of Trigonometric Functions
Section 5.3 - Inverse Functions
  • Definition of Inverse Function
  • Verifying Inverse Functions
  • Reflective Property of Inverse Functions
  • Existence of an Inverse Function
  • Guidelines for Finding an Inverse Function
  • Testing Whether a Function is One-to-One
  • Derivative of an Inverse Function
  • Continuity and Differentiability of Inverse Functions
  • Evaluating the Derivative of an Inverse Function
Section 5.4 - Exponential Functions: Differentiation and Integration
  • Definition of the Natural Exponential Function
  • Solving Exponential Equations
  • Solving Logarithmic Equations
  • Operations with Exponential Functions
  • Properties of Natural Exponential Functions
  • Derivative of the natural Exponential Function
  • Differentiating Exponential Functions
  • Locating a Relative Extrema
  • The Standard Normal Probability Density Functions
  • Integration Exponential Functions
  • Finding Areas Bounded by Exponential Functions
Section 5.5 - Bases Other Than e and Applications
  • Definition of Exponential Functions to Base e
  • Properties of Inverse Functions
  • Derivatives of Exponential Functions to Base e
  • Integrating Exponential Functions to Base e
  • The Power Rule for Real Exponents
  • Continuing Compounding and Annual Compounding
Section 5.6 - Inverse Trigonometric Functions: Differentiation
  • Definition of Inverse Trigonometric Functions
  • Evaluating Inverse Trigonometric Functions
  • Properties of Inverse Trigonometric Functions
  • Derivatives of Inverse Trigonometric Functions
  • Analyzing an Inverse Trigonometric Graph
Section 5.7 - Inverse Trigonometric Functions: Integration
  • Integrate Functions Whose Antiderivatives Involve Inverse Trigonometric Functions
  • Integration by Substitution
  • Use the Method of Completing the Square to Integrate a Function
Section 5.8 - Hyperbolic Functions
  • Definition of Hyperbolic Functions
  • Differentiation and Integration of Hyperbolic Functions
  • Finding Relative Extrema
  • Inverse Hyperbolic Functions
  • Differentiation and Integration of Inverse Hyperbolic Functions