## Course Content

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

### Math 1411A | Math 1411B | Math 1411C

##### Chapters 1 and 2

Section 1.1 - A Preview of Calculus
Section 1.2 - Finding Limits Graphically and Numerically
Section 1.3 - Evaluating Limits Analytically
Section 1.4 - Continuity and One-Side Limits
Section 1.5 - Infinite Limits
Section 2.1 - The Derivative and the Tangent Line Problem
Section 2.2 - Basic Differentiation Rules and Rates of Change
Section 2.3 - Product and Quotient Rules and Higher-Order Derivatives
Section 2.4 - The Chain Rule
Section 2.5 - Implicit Differentiation
Section 2.6 - Related Rates

- What is Calculus?
- The Tangent Line Problem
- The Area Problem

- 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

- 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

- 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

- Infinite Limits
- Definition of Infinite Limits
- Determining Infinite Limits from a Graph
- Vertical Asymptotes
- Properties of Infinite Limits
- Determining Infinite Limits

- 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

- 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

- 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

- 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

- 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

- Find a Related Rate
- Use Related Rates to Solve Real-Life Problems

### Math 1411B | Math 1411A | Math 1411C

##### Chapters 3, Sections 4.1 and 4.2

Section 3.1 - Extrema on an Interval
Section 3.2 - Rolle's Theorem and the Mean Value Theorem
Section 3.3 - Increasing and Decreasing Functions and the First Derivative Test
Section 3.4 - Concavity and the Second Derivative Test
Section 3.5 - Limits at Infinity
Section 3.6 - A Summary of Curve Sketching
Section 3.7 - Optimization Problems
Section 3.8 - Newton's Method
Section 3.9 - Differentials
Section 4.1 - Antiderivatives and Indefinite Integration
Section 4.2 - Area

- 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

- 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

- 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

- 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

- 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

- 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

- Solve Applied Minimum and Maximum Problems
- Guidelines for Solving Applied Minimum and Maximum Problems

- 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

- 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

- 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

- 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

### Math 1411C| Math 1411A | Math 1411B

##### Sections 4.3, 4.4, 4.5, 4.6, and Chapter 5 and 6

Section 4.3 - Riemann Sums and Definite Integrals
Section 4.4 - The Fundamental Theorem of Calculus
Section 4.5 - Integration by Substitution
Section 4.6 - Numerical Integration
Section 5.1 - The Natural Logarithmic Function: Differentiation
Section 5.2 - The Natural Logarithmic Function: Integration
Section 5.3 - Inverse Functions
Section 5.4 - Exponential Functions: Differentiation and Integration
Section 5.5 - Bases Other Than e and Applications
Section 5.6 - Inverse Trigonometric Functions: Differentiation
Section 5.7 - Inverse Trigonometric Functions: Integration
Section 5.8 - Hyperbolic Functions

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- Definition of Inverse Trigonometric Functions
- Evaluating Inverse Trigonometric Functions
- Properties of Inverse Trigonometric Functions
- Derivatives of Inverse Trigonometric Functions
- Analyzing an Inverse Trigonometric Graph

- Integrate Functions Whose Antiderivatives Involve Inverse Trigonometric Functions
- Integration by Substitution
- Use the Method of Completing the Square to Integrate a Function

- Definition of Hyperbolic Functions
- Differentiation and Integration of Hyperbolic Functions
- Finding Relative Extrema
- Inverse Hyperbolic Functions
- Differentiation and Integration of Inverse Hyperbolic Functions