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Calculus Series . Math Thought Program

Calculus 1

Differential Calculus Overview

A brief review of inequalities, functions and plane analytic geometry; limits and continuity; the derivative and the differential; applications of differentiation; L’Hospital’s Rule; introduction to the Riemann integral. Includes differentiation of logarithmic and exponential functions, and indeterminate forms. History of selected topics is studied. Four hours of lecture and one hour of laboratory/recitation. By the end of the course, students will be able to demonstrate an adequate understanding of these topics.

To be successful in this course students must be proficient in precalculus. Students may forgo enrolling in a precalculus course provided they have a strong background in college algebra and trigonometry. Please review the prerequisite before enrolling in this course. Students will not be able to complete the course without having a strong background in these subjects.

Prerequisite:  Precalculus  or  College Algebra and Trigonometry
Self-Paced
Study Time: 8hr/wk
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Video Lessons
20 Videos
📚
Topic Sections
4 Sections
🕑
Avg Per Lesson
~20 Minutes
📅
Duration
Self-paced
✍
Practice Sets
4+ Problems
🎓
Certificate
On Completion
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What You Need First
1
Foundations
Precalculus
A thorough review and strengthening of the algebraic and trigonometric foundations required for calculus. Covers functions, transformations, polynomial and rational...
Skills you will learn
  • Analyze and graph polynomial, rational, exponential, and logarithmic functions
  • Apply function transformations: shifts, reflections, stretches, compressions
  • Solve equations involving exponents, logarithms, and trigonometric functions
  • Evaluate trigonometric functions and apply identities
  • Graph trigonometric functions and solve trigonometric equations
~ Precalculus
View course ›
2
Alternative Path
College Algebra
Functions, equations, graphing, polynomial and rational expressions. Take this with Trigonometry if you did not take Precalculus.
Skills you will learn
  • Function notation and evaluation: f(x), f(a+h)
  • Factoring polynomials and rational expressions
  • Solving equations: polynomial, rational, radical
  • Graphing functions and transformations
  • Domain and range analysis
Why it matters
  • Every calculus technique requires algebraic manipulation
  • Limits require simplifying complex fractions and radicals
  • Optimization problems depend on setting up and solving equations
~ College Algebra
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3
Alternative Path
Trigonometry
Trigonometric functions, identities, equations, and applications. Required alongside College Algebra if you did not take Precalculus.
Skills you will learn
  • Evaluate trigonometric functions for any angle
  • Apply fundamental and Pythagorean identities
  • Solve trigonometric equations
  • Graph sine, cosine, tangent and their transformations
  • Understand inverse trigonometric functions
Why it matters
  • Derivatives of trig functions appear throughout Calculus 1
  • Integration techniques in Calculus 2 depend on trig identities
  • Related rates and optimization problems use trigonometric models
~ Trigonometry
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Course Topics & Lessons
01
[Cal1] (01) Limits
20 lessons
01.1
Single Variable Continuity
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01.2
Lesson 07 - Squeeze Theorem
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01.3
A Mindset for Success
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01.4
Fundamentals of Limits
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01.5
Precalculus Review
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01.6
Lesson 01 - Understanding Limits Numerically
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01.7
Lesson 08 - Advanced Limit
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01.8
Types of Discontinuities
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01.9
Calculus Controversy
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01.10
Lesson 05 - Overview of Continuity
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01.11
[APP] Graphical Limits
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01.12
Squeeze Theorem
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01.13
Lesson 06 - Verifying Continuity
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01.14
What is Calculus?
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01.15
Understanding Limits
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01.16
Lesson 02 - Infinite Limits
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01.17
The Intermediate Value Theorem (IVT)
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01.18
Continuity of Functions
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01.19
Lesson 03 - Limit Properties
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01.20
Understanding Indeterminate Forms
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01.21
[APP] Graphical Infinite Limits
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01.22
Lesson 04 - Indeterminate Forms
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01.23
Infinite Limits and Limit Properties
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01.24
A Preview of Precision: Delta–Epsilon Limits
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02
[Cal1] (02) Derivatives
30 lessons
02.1
Why Study Derivatives?
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02.2
Foundational Rules
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02.3
Trigonometric Functions
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02.4
Exponentials and Logarithms
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02.5
Derivatives of Inverses
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02.6
Rates of Change
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02.7
Lesson 18 - Derivatives Log & Inverse Trig
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02.8
Lesson 15 - Derivates of Inverses
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02.9
Inverse Trigonometric Functions
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02.10
Operational Rules
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02.11
[Flash Cards] Properties of Exponents
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02.12
Lesson 19 - Inverse & Logarithmic Differentiation
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02.13
Instantaneous Rate of Change
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02.14
Lesson 11 - Power Rule
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02.15
Lesson 16 - Inverse Trig Derivatives
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02.16
Lesson 14 - Trigonometric Derivatives
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02.17
[Flash Cards] Properties of Logarithms
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02.18
Lesson 09 - Rates of Change
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02.19
Lesson 12 - Product & Quotient Rule
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02.20
[Flash Cards] Derivative Rules
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02.21
Implicit Differentiation
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02.22
[Flash Cards] Basic Derivative Formulas
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02.23
Lesson 17 - Implicit Differentiation
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02.24
[APP] Rates of Change
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02.25
Chain Rule
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02.26
[Flash Cards] Trigonometric Derivative Formulas
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02.27
Check Your Understanding
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02.28
Lesson 13 - Chain Rule
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02.29
Lesson 10 - The Derivative Function
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02.30
Lesson 13b - Chain Rule
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02.31
Derivatives Limit Definition
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03
[Cal1] (03) Applications of Derivatives
74 lessons
03.1
Introduction
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03.2
Introduction
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03.3
Introduction
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03.4
Overview
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03.5
Introduction
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03.6
Introduction
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03.7
Introduction
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03.8
Introduction
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03.9
Study Playbook
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03.10
Rolle's Theorem
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03.11
Example: Understanding Motion
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03.12
Increasing and Decreasing Functions
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03.13
Lecture
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03.14
Process Overview: Optimization
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03.15
Geometry Formulas
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03.16
Concavity and Inflection Points
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03.17
Learning Objectives
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03.18
Test Your Might: Exam Prep
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03.19
Example: Rolle's Theorem
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03.20
Practice Problem: Components of Motion
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03.21
Process Overview
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03.22
Definitions for Increasing & Decreasing
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03.23
Second Derivative Test
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03.24
Handout - Optimization
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03.25
How To Use
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03.26
Key Identities
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03.27
Potential Pitfalls
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03.28
Linearization Error Analysis
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03.29
Lecture: Optimization [Part 01]
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03.30
Definitions for Monotonicity
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03.31
Lecture: MVT
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03.32
Using the Second Derivative
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03.33
Lecture: Curve Sketching [Understanding The Derivative]
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03.34
Handout: Related Rate
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03.35
Mean Value Theorem
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03.36
Lecture: Optimization [Part 02]
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03.37
Linearization vs. Error Analysis
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03.38
Sign Chart Method
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03.39
Example: Mean Value Theorem
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03.40
Lecture - Part 01
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03.41
Example: Optimization
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03.42
Using the First Derivative
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03.43
APP: Linearization & Differentials
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03.44
Example: Using the Second Derivative
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03.45
APP: Linearization Error Analysis
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03.46
Rolle's Theorem vs. MVT
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03.47
Example: SDT & Classify Extrema
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03.48
Math In Motion: Wire Cut
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03.49
Lecture - Part 02
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03.50
Sign Chart Method
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03.51
Key Concepts
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03.52
Math In Motion: Spherical Balloon
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03.53
Math In Motion: Norman Window
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03.54
Example: Using the First Derivative
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03.55
Example: Approximating Sine
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03.56
Practice Problems: Linearization
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03.57
1st Derivative Test or 2nd Derivative Test
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03.58
When the MVT Fails
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03.59
Math In Motion: Conical Tank
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03.60
Practice Problems: Optimization
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03.61
Critical Points & Critical Values
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03.62
Practice Problems: Rolle's Theorem & MVT
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03.63
Math In Motion: Sliding Ladder
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03.64
Local and Absolute Extrema
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03.65
Math Byte: Curve Sketching with Calculus
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03.66
Math In Motion: Hot Air Balloon
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03.67
Practice Problems: Concavity
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03.68
Lecture: Relative Extrema
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03.69
Math In Motion: Baseball Diamond
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03.70
Practice Problems: Extrema
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03.71
Lecture: Absolute Extrema
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03.72
Math Bytes: Related Rates & Spheres
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03.73
Lecture: Curve Sketching Basics
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03.74
Math Byte: Curve Sketching with Calculus
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03.75
Practice Problems: Monotonicity
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04
[Cal1] (04) L'Hôpital's Rule
21 lessons
04.1
Live Lecture [Part 01]
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04.2
Introduction
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04.3
Math In Motion: Example 01
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04.4
Math In Motion: Example 01
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04.5
Math In Motion: Example 01
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04.6
Math In Motion: Example 01
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04.7
Math In Motion: Example 01
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04.8
Math In Motion: Example 01
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04.9
Math In Motion: Example 01
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04.10
Math In Motion: Example 01
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04.11
Live Lecture [Part 02]
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04.12
Math In Motion: Example 02
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04.13
Math In Motion: Example 02
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04.14
Math In Motion: Example 02
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04.15
Math In Motion: Example 02
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04.16
Flashcards
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04.17
Test Your Might!!
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04.18
Math In Motion: Example 03
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04.19
Math In Motion: Example 03
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04.20
L'Hôpital's Rule
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04.21
Practice Assignments
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04.22
Math In Motion: Example 04
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04.23
Compute the following limits
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05
[Cal1] (05) Integration Basics
13 lessons
05.1
Lecture: Introduction to Riemann Sums
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05.2
Lecture: Fundamental Theorem of Calculus
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05.3
Introduction
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05.4
Lecture: The Second FTC and Average Function Value
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05.5
Lecture: Computing Areas By Rectangles
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05.6
Basic Integration Identities
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05.7
Lecture: U-Substitution [Part 01]
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05.8
Summation Flashcards
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05.9
Integral and Differentiation Notation
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05.10
Lecture: U-Substitution [Part 02]
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05.11
Lecture: Antiderivatives [Part 01]
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05.12
Lecture: Computing More Areas By Riemann Sums
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05.13
Assignments: Riemann Sums
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05.14
Lecture: Antiderivatives [Part 02]
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05.15
Practice Problems
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06
[Cal1] (06) Applications of Integrals
5 lessons
06.1
Lecture: Area Between Curves
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06.2
Lecture: Volumes
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06.3
Lecture: Arc Length
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06.4
Lecture: Disk Method
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06.5
Lecture: Shell Method
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Course Materials
● Required 2 books
Course Textbook
Thomas Calculus: Early Transcendentals
Joel Hass, Christopher Heil, Przemyslaw Bogacki
2023 Pearson Education ISBN: 9780137728626
Course Textbook
MyLab Math eText
18 Weeks Pearson ISBN: 9780137559794
How This Helps You
Mathematics
How Mathematics uses this course
Where you will use this directly
Differential Equations
You'll master derivatives and understand how functions behave as they change. In Differential Equations, you'll model everything from population growth to heat flow using these same derivative concepts. This builds your ability to see mathematical patterns across different contexts.
Linear Algebra
You'll learn to solve systems of equations and work with matrices in Calculus 1. This foundation is essential for computer graphics, data analysis, and machine learning applications. These problem-solving skills build your confidence in tackling complex, multi-step challenges.
Economics
You'll learn to analyze marginal cost and revenue functions using calculus principles. This mathematical foundation helps you understand market behavior and optimize business strategies. The analytical mindset you develop applies to all decision-making processes.
Computer Science
You'll practice breaking down complex problems into manageable steps through function analysis. This systematic thinking is essential for programming, algorithm design, and software development careers.
Computer Science
How Computer Science uses this course
Where you will use this directly
Machine Learning
You'll use derivatives to optimize neural networks and minimize error functions in gradient descent algorithms.
Computer Graphics
Calculus helps you render smooth curves, calculate lighting models, and create realistic 3D transformations.
Algorithms & Data Structures
Analyze algorithm complexity using limits and understand how functions grow as input size increases.
Game Development
Model physics simulations, particle effects, and smooth character movement using calculus principles.

Instructor information will be available once you are enrolled.

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Calculus 2
Integral Calculus