Single Variable Calculus

This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics. Course Format This course has been designed for independent study. It includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include: - **Lecture Videos** with supporting written notes - **Recitation Videos** of problem-solving tips - **Worked Examples** with detailed solutions to sample problems - **Problem sets** with solutions - **Exams** with solutions - **Interactive Java Applets** ("Mathlets") to reinforce key concepts Content Development David Jerison    Arthur Mattuck    Haynes Miller    Benjamin Brubaker    Jeremy Orloff Heidi Burgiel    Christine Breiner    David Jordan    Joel Lewis About OCW Scholar OCW Scholar courses are designed specifically for OCW's single largest audience: independent learners. These courses are substantially more complete than typical OCW courses, and include new custom-created content as well as materials repurposed from previously published courses.

• Clip: Finale
• Clip 1: Alternate Definition of Natural Log
• Clip 1: Antiderivative of 1/x
• Clip 1: Area of Part of a Circle
• Clip 1: Area Under the Bell Curve
• Clip 1: Areas Between Curves
• Clip 1: By Horizontal Slices
• Clip 1: Comparison of the Harmonic Series
• Clip 1: Completing the Square
• Clip 1: Curves are Hard, Lines are Easy
• Clip 1: Derivative of ax
• Clip 1: Derivative of sin(x), Algebraic Proof
• Clip 1: Difference Quotient of ax
• Clip 1: Differential Equations and Slope, Part 1
• Clip 1: Differentiation Formulas
• Clip 1: dy/dx=f(x)
• Clip 1: Example 1: y=1/x
• Clip 1: Example of Estimation
• Clip 1: Example: Cumulative Debts
• Clip 1: Functions Without Elementary Anti-Derivatives
• Clip 1: General Strategy for Graph Sketching
• Clip 1: Higher Derivatives
• Clip 1: Indefinite Integrals over Singularities
• Clip 1: Integral of 1/(xp)
• Clip 1: Integral of cos2(x)
• Clip 1: Introduction of Product and Quotient Rules
• Clip 1: Introduction to Arc Length
• Clip 1: Introduction to Comparison
• Clip 1: Introduction to Curve Sketching
• Clip 1: Introduction to Definite Integrals
• Clip 1: Introduction to Differentiation
• Clip 1: Introduction to Implicit Differentiation
• Clip 1: Introduction to Improper Integrals
• Clip 1: Introduction to L'Hospital's Rule
• Clip 1: Introduction to Newton's Method
• Clip 1: Introduction to Numerical Integration
• Clip 1: Introduction to Ordinary Differential
• Clip 1: Introduction to Polar Coordinates
• Clip 1: Introduction to Power Series
• Clip 1: Introduction to Rates of Change
• Clip 1: Introduction to Related Rates
• Clip 1: L'Hospital's Rule, Continued
• Clip 1: List of Approximations
• Clip 1: Maxima and Minima Using Graphs
• Clip 1: Minimal Surface Area of a Box: Direct Solution
• Clip 1: Non-Constant Speed Parametrization
• Clip 1: Other Bases
• Clip 1: Partial Fractions I
• Clip 1: Probability Example
• Clip 1: Probability Summary
• Clip 1: Proof of the Second Fundamental Theorem of Calculus
• Clip 1: Questions on Exam III
• Clip 1: Quotient Rule
• Clip 1: Remarks on Notation
• Clip 1: Review for Test IV
• Clip 1: Review of Taylor's Series
• Clip 1: Review of the Fundamental Theorem
• Clip 1: Review of Trigonometric Identities
• Clip 1: Substitution
• Clip 1: Substitution When u' Changes Sign
• Clip 1: The Bell Curve
• Clip 1: The Mean Value Theorem and Linear Approximation
• Clip 1: The Most Natural Logarithmic Function
• Clip 1: Value of e
• Clip 1: Volume of a Cauldron
• Clip 1: Volume of a Sphere
• Clip 1: Why L'Hospital's Rule Works
• Clip 2: Antiderivative of xa
• Clip 2: Arclength of Parametric Curves
• Clip 2: Area of an Off-Center Circle
• Clip 2: Area Under the Bell Curve
• Clip 2: Boiling Cauldron: Introduction
• Clip 2: By Vertical Slices
• Clip 2: Comparison Example
• Clip 2: Comparison Tests
• Clip 2: Consequences of the Mean Value Theorem
• Clip 2: Continuity
• Clip 2: Derivative of cos(x)
• Clip 2: Differential Equations and Slope, Part 2
• Clip 2: Divergent Series
• Clip 2: Example: Change of Variables
• Clip 2: Example: Dnxn
• Clip 2: Example: y=mx
• Clip 2: Explaining the Formula by Example
• Clip 2: Exponent Review
• Clip 2: General Power Series
• Clip 2: Geometric Interpretation of Differentiation
• Clip 2: Graph of r = sin(2θ)
• Clip 2: Improper Integrals of the Second Kind, Continued
• Clip 2: Integral of ln(x)2
• Clip 2: Integral of Tangent
• Clip 2: Integration by "Advanced Guessing"
• Clip 2: Introduction to General Rules for Differentiation
• Clip 2: Limit of (1-cos(x))/x
• Clip 2: Limit of sin(x)/(x2)
• Clip 2: Log of a Product
• Clip 2: Maximum Area of Two Squares
• Clip 2: Minimal Surface Area of a Box: Implicit Differentiation
• Clip 2: Proof of the First Fundamental Theorem of Calculus
• Clip 2: Quadratic Factors
• Clip 2: Separation of Variables
• Clip 2: Simple Examples in Polar Coordinates
• Clip 2: Solids of Revolution
• Clip 2: Stacking Blocks
• Clip 2: Taylor's Series of 1/(1 + x)
• Clip 2: The Fundamental Theorem and the Mean Value Theorem
• Clip 2: The Mean Value Theorem and Inequalities
• Clip 2: Types of Riemann Sums
• Clip 2: Using the First Fundamental Theorem
• Clip 2: Using the Second Fundamental Theorem of Calculus
• Clip 3: Asymptotes of Antiderivatives
• Clip 3: Average Height with Respect to Arc Length
• Clip 3: Boiling Cauldron, Continued
• Clip 3: Derivative of xx
• Clip 3: Example: Boy Near a Dart Board
• Clip 3: Example: y4+xy2-2=0
• Clip 3: Examples of Comparison
• Clip 3: Graphing a Rational Function
• Clip 3: Infinite Discontinuities
• Clip 3: Integral of e-x2
• Clip 3: Linear Approximation and the Definition of the Derivative
• Clip 3: Newton's Method: What Could Go Wrong?
• Clip 3: Notation for Derivatives
• Clip 3: Notation for Series
• Clip 3: Numerical Integration Study Tips
• Clip 3: Polar Coordinates and Conic Sections
• Clip 3: Product Rule
• Clip 3: Question: Can We Use the Original Formula?
• Clip 3: The Cover-Up Method
• Clip 3: Translating y = 1 into Polar Coordinates
• Clip 3: Trapezoidal Rule
• Clip 4: An Improper Integral of the Second Kind
• Clip 4: Another Reduction Formula
• Clip 4: d/dx (ln(sec x))
• Clip 4: Example 2: y=xn
• Clip 4: Examples of Series
• Clip 4: Integral of sin3(x)
• Clip 4: Other (Ugly) Discontinuities
• Clip 4: Physical Interpretation of Derivatives, Continued
• Clip 4: Slope as Ratio
• Clip 4: Substitution of Power Series
• Clip 4: Summary of Trig Integration
• Clip 6: The Derivative of Everything?
• Clip 7: Key Concepts in Differentiation