Larry Green's Calculus Videos

Limits    Derivatives    Applications of the Derivative    Antiderivatives and Definite Integration

Logs, Exponents and Transcendental Functions    Differential Equations    Applications of Integration

Integration Techniques, L'Hopital's Rule, and Improper Integrals     Sequences and Series

Conics, Parametric Equations, and Polar Equations    Vectors    Functions of Several Variables   

Vector-Valued Functions    Multiple Integration    Vector Calculus

This collection of videos is intended as a resource for working out typical problems that are assigned in the first four quarters or three semesters of calculus.  It is assumed that the student has attended lecture to get the theory and equations that are introduced in every calculus course. 

Larry Green's Calculus Videos are Now Available as Apps for the iPhone:

• App for Calculus I (First Quarter Calculus)

• App for Calculus II (Second Quarter Calculus)

• App for Calculus III (Third Quarter Calculus)

• App for Calculus IV (Fourth Quarter Calculus)

Limits:

• A Preview of Calculus
  • Using the Secant Line to Approximate the Tangent Line
    
    
• Limits Graphically and Numerically
  • Using Tables to Approximate Limits
  • Using the Graph to Approximate a Limit
  • Using the Graph to See that a Limit Does Not Exist
  • Finding d Given e with the e-d Definition of the Limit
  • Using the e-d Definition of the Limit
    
    
• Limits Analytically
  • Using Factoring to Find a Limit
  • Using Conjugates Roots to find a Limit
  • A Limit that Involves sin x
    
    
• Continuity and One Sided Limits
  • Limits, Continuity, and Graphs
  • Finding Removable and Non-Removable Discontinuities
  • Finding a One Sided Limit
  • Continuity and Piecewise Functions
  • Using the Intermediate Value Theorem to Show a Function has a zero
    
    
• Asymptotes
  • One Sided Limits and Asymptotes

Derivatives 

• Limit Definition of the Derivative
  • Using the Limit Definition to Find the Derivative of a Rational Function
  • Using the Limit Definition to Find the Derivative of a Function that Involves Roots
  • Using the Limit Definition to Find the Equation of the Tangent Line
  • Sketching the Graph of f '(x) Given the Graph of f(x) (Piecewise Linear Function)
  • Sketching the Graph of f '(x) Given the Graph of f(x) (A more curvy curve)
    
    
• Basic Derivative Rules
  • Using the Power Rule to Find a Derivative
  • Using the Derivative Rules to Find the Equation of the Tangent Line
  • Using Algebra and the Power Rule to Find the Derivative
  • Finding Impact Velocity
    
    
• Product and Quotient Rules
  • Using the Product Rule for Derivatives
  • Using the Quotient Rule for Derivatives
  • Finding a Second Derivative
    
    
• The Chain Rule
  • Using the Chain Rule for Derivatives
  • Using the Chain and Product Rules Together
  • Using the Chain Rule to Find the Equation of the Tangent Line
    
    
• Implicit Differentiation
  • Using Implicit Differentiation to Find a Derivative
  • Using Implicit Differentiation to Find the Equation of a Tangent Line
  • Using Implicit Differentiation to Find the Second Derivative

Applications of the Derivative

• Related Rates
  • Related Rates:  An Expanding Balloon
  • Related Rates:  Changing the Angle of a Camera
  • Related Rates:  Finding the Change of a Shadow Length
  • Related Rates:  How Fast are Two Airplanes Approaching Each Other?
    
    
• Critical Points and Extrema on an Interval
  • Finding Critical Points
  • Finding Absolute Extrema on a Closed Interval
    
    
• Rolle's Theorem and the Mean Value Theorem
  • Checking the Conditions of Rolle's Theorem
  • Find c Guaranteed by the Mean Value Theorem
  • Using Rolle's Theorem to Show Velocities are Equal
  • Proof of the Fixed Point Theorem Using the Mean Value Theorem
    
    
• Increasing and Decreasing Functions and the First Derivative Test
  • Determine When a Function is Increasing or Decreasing
  • Apply the First Derivative Test to Locate and Classify Relative Extrema
    
    
• Concavity and the Second Derivative Test
  • Finding Inflection Points and discussing Convavity
  • Using the Second Derivative Test
    
    
• Limits at Infinity
  • Finding a Limit as x Approaches Infinity
  • Finding a Limit as x Approaches Negative Infinity
  • Using the e-M Definition to Prove a Limit is 0
    
    
• Calculus and Curve Sketching
  • Sketching the Graph of a Polynomial
  • Sketching a Graph With a Vertical Asymptote
  • Sketching a Graph With Both Vertical Asymptotes and Horizontal Asymptotes
    
    
• Optimization
  • Finding the Minimum Surface Area of a Can
  • Minimizing Transit Time
  • Finding the Rectangle With Maximum Area That is Bounded by a Curve
    
    
• Newton's Method
  • Using Newton's Method to Approximate the Intersection of 2 Curves
  • An Example Where Newton's Method Fails
    
    
• Differentials
  • Differentials and Comparing Dy and dy
  • Using Differentials to Approximate a Square Root
  • Using Differentials Graphically to Approximate a Function Value
  • Using Differentials to Approximate Percent Propagated Error
  • Using Differentials to Prove Relative Errors are Equal in Magnitude

Antiderivatives and Indefinite Integration

• Antiderivatives
  • Using the Power Rule to Integrate
  • Solving a Differential Equation by Integrating
  • Using Acceleration to Find a Maximum Height
    
    
• Area
  • Using Right Rectangles to Approximate Area
  • Using Left Rectangles to Approximate Area
  • Using the Limit Definition to Find Area
    
    
• The Definite Integral
  • Looking at a Graph and Setting Up a Definite Integral
  • Using Geometry to Evaluate a Definite Integral
  • Using the Properties of the Definite Integral
  • Integrating a Piecewise Linear Function Using Its Graph
    
    
• The Fundamental Theorem of Calculus
  • Using the 1^st Fundamental Theorem of Calculus   
  • Finding Area of a Region   
  • Finding the Average Value   
  • Using the 2^nd Fundamental Theorem of Calculus   
  • The Chain Rule and the 2^nd Fundamental Theorem  
  • Integration Props and the 2^nd FTC
  • Sketching the Graph of y = f (x) Given the Graph of y = f '(x)
    
    
• Substitution
  • u-substitution   
  • u-substitution and Working with Constants   
  • A More Complicated Substitution
    
    
• Numerical Integration
  • Using the Trapezoidal Rule to Approximate an Integral
  • Using Simpson's Rule to Approximate an Integral
  • Finding the Maximum Error in Simpson's Rule
  • Find n to Approximate an Integral with the Trapezoidal Rule

Logs, Exponents and Transcendental Functions

• Logs and Derivatives
  • Finding the Derivative of a ln Function   
  • Logarithmic Differentiation
    
    
• Logs and Integrals
  • Integrating With Ln   
  • Advanced u-substitution and Ln
    
    
• Inverse Functions
  • Showing a Function has an Inverse   
  • Finding the Derivative of the Inverse 
  • Finding the Derivative of the Inverse Using the 2^nd Fundamental Theorem
    
    
• Exponential Functions
  • The Derivative of e^x   
  • Solving a Differential Equation Involving e^x
    
    
• Other Bases
  •  The Derivative of an Exponential with a Non-constant Base   
  • Tangent Line with a Common Log    
  • Integration with Other Bases
    
    
• Derivatives of Inverse Trig Functions
  • Derivatives Involving arcsin x   
  • Derivatives Involving arctan x   
  • Derivatives Involving arcsec x
    
    
• Integrals Involving Inverse Trig Functions
  • Integrals Involving arcsin x   
  • Integrals Involving arctan x   
  • Integrals Involving arcsec x
    
    
• Hyperbolic Trig Functions
  • Integrals that Contain Hyperbolic Trig      
  • Using Inverse Hyperbolic Trig for Integration

Differential Equations

• Slope Fields and Euler's Method
  • Matching Slope Fields to Differential Equations 
  •  Verifying an Equation is a Solution to a Differential Equation 
  • Using Euler's Method
    
    
• Growth and Decay
• Exponential Growth  
• Finding Age using Half-Life  
• Newton's Law of Cooling
  
  
• Separation of Variables and the Logistic Equation
•  Separation of Variables   
• Homogeneous Differential Equations  
• Logistics Equations

Applications of Integration

• Area Between Two Curves
  • Finding the Area Between 2 Curves
    
    
• Volume by Disks and Washers and Cross-Sections
  • How to Use the Washer Method  
  • Revolving About a Line Below the x-axis  
  • Revolving About a Line Above the x-axis 
  • Volumes by Cross Section
    
    
• Volumes by Cylindrical Shells
  • Using the Method of Cylindrical Shells  
  • Revolving About a Line Other than the y-axis
    
    
• Arc Length and Surface Area
  • Finding Arc Length   
  • Surface of Revolution About the y-axis  
  • Surface of Revolution About the x-axis
    
    
• Work
  • Using Hooke's Law 
  • Work to Send a Satellite Into Orbit  
  • Work Done Lifting a Chain  
  • Work Done Emptying a Tank    
    
    
• Center of Mass
  • Finding the Centroid  
  • Using the Theorem of Pappus
    
    
• Fluid Pressure
  • Finding the Fluid Pressure on a Vertical Triangle   
  • Finding the Fluid Pressure on a Vertical Semi-Circle

Integration Techniques, L'Hopital's Rule and Improper Integrals

• Integration by Parts
  • Integration by Parts  
  • Integrating Arcsin(x) 
  • Integrating e^x cos(x)
    
    
• Integrating Powers of Trig Functions
  • Integrating cos⁵(x) 
  • Integrating sin⁴(x)   
  • Integrating sec x tan³x   
    
    
• Arc Trig Substitution
  • ArcSin Substitution 
  • ArcTan Substitution  
  • ArcSec Substitution
    
    
• Partial Fractions
  • Partial Fractions (Simple) 
  • Partial Fraction (One Factor is a Quadratic)  
  • Partial Fractions (Repeated Roots)
    
    
• L'Hopital's Rule
  • Applying L'Hopital's Rule Twice   
  • L'Hopital's Rule With Exponents 
  • L'Hopital's Rule of the Form 0 Times Infinity   
    
    
• Improper Integrals
  • Improper Integrals (Upper Bound is Infinity) 
  • Improper Integrals (With an Asymptote)

Sequences and Series

• Sequences
  • Simplifying a Quotient Involving Factorials
  • Determining if a Sequence Converges by Taking a Limit
  • Determining if a Sequence that Involves Factorials Converges by Taking a Limit
    
    
• Series
  • Using the n^th Term Test to Show that a Series Diverges
  • Finding the Sum of a Telescoping Series
  • Determining if a Geometric Series Converges and then Finding what it Converges to
  • Find the Values of x that Make a Geometric Series Converge
    
    
• The Integral Test
  • Using the Integral Test to Show that a Series Converges
    
  • Using the Integral Test to Show that a Series Diverges
  • Using the p-Series Test to Show that a Series Diverges.
    
    
• The Comparison Tests
  • Using the Direct Comparison Test to Show that a Series Converges
  • Using the Direct Comparison Test to Show that a Series Diverges
  • Using the Limit Comparison Test to Show that a Series Converges
  • Using the Limit Comparison Test to Show that a Series Diverges
    
    
• Alternating Series
  • Showing that an Alternating Series Converges Absolutely
  • Showing that an Alternating Series Diverges
  • Showing that an Alternating Series Converges Conditionally
  • Finding N such that the Error in Approximating a Series is Less Than 0.001
  
  
• The Ratio and Root Tests
  • Using the Ratio Test to Show that a Series Converges
  • Using the Ratio Test to Show that a Series Diverges
  • Shifting the Index of Summation from n=2 to n=0
  • Using the Root Test to Show that a Series Converges
  • Using the Root Test to Show that a Series Diverges
    
    
• Taylor and Mclaurin Polynomials
  • Finding a Mclaurin Polynomial
  • Finding a Third Degree Taylor Polynomial and Using it to Approximate a Value
  • Using Taylor's Remainder Theorem to Find an Upper Bound for the Error
  • Determining the degree of the Mclaurin polynomial needed to approximate e.
    
    
• Power Series
  • Finding the Interval of Convergence
  • Finding the Interval of Convergence (involves factorials and an infinite radius of convergence)
  • Finding the Interval of Convergence (involves (3)(5)(7)...(2n-1) and a zero radius of convergence)
  • Verifying that a Power Series is a Solution to a Differential Equation
    
    
• Geometric Power Series
  • Writing a Function as a Geometric Power Series Centered at c = 0
  • Writing a Function as a Geometric Power Series Centered at c = 3
  • Using a Geometric Power Series to Approximate an Integral
  • Using a Well-known Function's Power Series to Evaluate a Power Series
    
    
• Taylor and Mclaurin Series
  • Use the Definition to Find the Taylor Series of a Function Centered at c = -2
  • Use the Binomial Series Formula to Find a Mclaurin Series
  • Using the Known Mclaurin Series for cos x to Find the Mclaurin Series for a Related Function
  • Finding a Mclaurin Series of a Product by Multiplying Two Mclaurin Series Together
  • Use a Mclaurin Series to Find a Limit
  • Use a Mclaurin Series to Find an Integral that Represents a Probability (Normal Distribution)

Conics, Parametric Equations, and Polar Equations

• Conics and Calculus
  • Finding the Equation of the Tangent Line to the Hyperbola
  • Finding the Area of an Ellipse
    
    
• Parametric Equations
  • Finding Parametric Equations for y = f(x)
  • Finding Parametric Equations of a Line
  • Finding Parametric Equations of a Circle
  • Eliminating the Parameter and using the Pythagorean Trig Identity
  • Eliminating the Parameter to Sketch a Graph
    
    
    
• Parametric Equations and Calculus
  • Finding the First and Second Derivative for Parametric Equations
  • Finding the Equation of the Tangent Lines Where a Curve Intersects Itself
  • Finding Points Where There are Horizontal and Vertical Tangent Lines
  • Finding the Arc Length of a Curve
  • Finding the Surface Area of Revolution about the x-axis and the y-axis
    
    
• Polar Coordinates
  • Converting From a Rectangular Equation to Polar Form
  • Converting From a Polar Equation to Rectangular Form
  • Finding the Slope of the Tangent Line (Polar Form)
  • Find the Horizontal and Vertical Tangent Lines to a Polar Graph
    
    
• Area and Arc Length in Polar Coordinates
  • Finding the Intersection of Two Polar Curves
  • Finding the Area of a Polar Region
  • Finding the Arc Length of a Polar Curve
  • Finding the Surface Area of Revolution of a Polar Curve

Vectors

• Plane Vectors
  • Finding the Unit Vector in the Direction of Another Vector
  • Vector Addition and Scalar Multiplication and the Geometric Interpretation
  • Using Vectors to Find the Airspeed of an Airplane
    
    
• 3 Dimensional Vectors
  • Plotting a Point in the xyz-Space
  • Determine if a Triangle is Right, Isosceles, or Neither
  • Determine if Three Points are Collinear
  • Describing a Solid that is the Outside of a Sphere
    
    
• The Dot Product
  • Finding the Angle Between Two Vectors
  • Finding Vectors that are Orthogonal to a Given Vector
  • Determining if a Triangle is Acute, Obtuse, or Right
  • Finding the Projection of u Onto v and the Component of u Orthogonal to v
  • Finding the Work Done on and Object
  • Proving the Diagonals of a Rhombus are Perpendicular
    
    
• The Cross Product
  • Finding the Area of a Parallelogram
  • Finding the Volume of a Parallelepiped
  • Calculating the Torque
    
    
• Lines and Planes
  • Finding the Equation of a Line that is Parallel to a Given Line
  • Finding the Point of Intersection of Two Lines
  • Finding the Equation of a Plane that Passes Through Three Given Points
  • Sketching a Plane
  • Finding the Distance Between a Point and a Plane
    
    
• Surfaces
  • Sketching a Cylinder
  • Sketching an Ellipsoid
  • Sketching a Paraboloid
  • Finding the Equation of a Surface of Revolution
    
    
• Cylindrical and Spherical Coordinates
  • Converting a Rectangular Equation to Cylindrical and Spherical Coordinates
  • Converting an Equation From Cylindrical Coordinates to Rectangular Coordinates
  • Converting an Equation From Spherical Coordinates to Rectangular Coordinates

Functions of Several Variables

• Introduction to Functions of Several Variables
  • Finding the Domain and Range of a Function of Two Variables
  • Sketching Level Curves for a Contour Diagram and Describing a Surface
    
    
• Limits
  • Showing a Limit Does Not Exist
  • Using Polar Coordinates to Find a Limit that Exists
  • Discussing the Continuity of a Function:  Removable Discontinuity
    
    
• Partial Derivatives
  • Finding the First Partial Derivatives of a Function
  • Using the Limit Definition to Find a Partial Derivative With Respect to y
  • Verifying that a Function is a Solution to a Partial Differential Equation (Laplace's Equation)
    
    
• Differentials
  • Approximating the Propagated Error
  • Using Differentials to Approximate the Percent Change
    
    
• The Chain Rule
  • The Chain Rule an Polar Coordinates
  • Using the Chain Rule for a Second Derivative
  • Differentiating a Function of Four Variables Implicitly
    
    
• Directional Derivatives and the Gradient
  • Finding the Directional Derivative
  • Using the Gradient Vector to Find a Normal Vector to a Level Curve
  • Finding the Path of a Heat Seeking Particle
    
    
• Tangent Planes and Normal Lines
  • Finding the Equation of the Tangent Plane to a Surface
  • Finding Symmetric Equations to the Normal Line to a Surface
  • Finding the Equation of the Tangent Line to the Curve of Intersection of Two Surfaces
    
    
• Relative and Absolute Extrema
  • Locating and Classifying Relative Extrema Using the Second Partials Test
  • Finding the Absolute Extrema Over a Triangle
    
    
• Applications of Relative Extrema
  • Finding the Dimensions that Maximize the Volume of a Rectangular Building
  • Using Least Squares Regression
    
    
• Lagrange Multipliers
  • Minimizing a Distance
  • Finding a Maximum (2 Variables)
  • Locating Extrema for a Function of 3 Variables Given 2 Constraints

Vector Valued Functions

• Vector Valued Functions (Definitions)
  • Finding the Domain of a Vector Valued Function
  • Sketching the Graph Represented by a Vector Valued Function (Helix)
  • Determining Whether Curves Intersect or Collide
    
    
• Derivatives and Integrals of Vector Valued Functions
  • Verifying a Property Involving Derivatives and Dot Products
  • Finding Where a Curve is Smooth
  • Finding an Indefinite Integral
  • Find r(t) Given r''(t) and r'(0) and r(0)
  • Proving a Property About Vector Valued Functions
    
    
• Velocity and Acceleration
  • Finding the Velocity and Acceleration
  • Finding the Minimal Angle Required to Hit a Home Run
  • Finding an Initial Speed and a Maximal Height
    
    
• Unit Tangent and Unit Normal Vectors
  • Finding the Velocity, Acceleration, Unit Tangent, and Unit Normal Vectors (2D)
  • Finding the Unit Tangent and Unit Normal Vectors an the Tangential and Normal Components of Acceleration (3D)
    
    
• Arc Length and Curvature
  • Finding the Arc Length of a Space Curve
  • Parameterizing a Curve by Arc Length
  • Finding the Curvature of a Curve Parameterized by Arc Length
  • Finding the Curvature of a Curve Given  by a General Vector Valued Function
  • Finding the Point Where a Curve has Maximum Curvature (y = e^x)

Multiple Integration

• Double Integrals and Area
  • Using a Double Integral to Find the Area Bounded by Two Curves
  • Switching the Order of Integration
  • Integrating by Switching the Order of Integration
  • Splitting the Region Into Two Regions and Using Two Double Integrals
    
    
• Double Integrals and Volume
  • Using a Formula From Geometry to Evaluate a Double Integral
  • Using a Double Integral to Find Volume
  • Showing a Function is a Joint Density Function and then Finding a Probability
    
    
• Double Integrals in Polar Coordinates
  • Using Polar Coordinates to Evaluate a Double Integral
  • Combining Two Messy Integrals into One Nice Integral in Polar Coordinates
  • Finding the Area that is Inside a Cardioid and Outside a Circle
    
    
• Center of Mass and Moment of Inertia
  • Finding the Center of Mass of a Lamina
  • Finding the Moments of Inertia and Radii of Gyration
  • Finding the Moment of Inertia of a Lamina About the Line x=2
    
    
• Surface Area
  • Finding the Surface Area Over a Triangle
  • Finding the Surface Area of the Part of a Sphere that Lies Over a Circle
    
    
• Triple Integrals
  • Sketching a Solid and Switching the Order of Integration
  • Using a Triple Integral to Find a Volume
  • Finding the Moment of Inertia (Assumes Constant Density 1)
    
    
• Triple Integrals in Cylindrical and Spherical Coordinates
  • Using Cylindrical Coordinates to Find the Mass of a Solid
  • Using Spherical Coordinates to Find the Volume in a Sphere and a Cone
    
    
• Jacobians
  • Using the Jacobian for a Parallelogram Region
  • Using the Jacobian for a Curvy Region
  • Finding the Volume of an Ellipsoid Using Jacobians

Vector Calculus

• Vector Fields
  • Sketching a Vector Field
  • Determining if Vector Fields are Conservative and Finding a Potential Function (Two Variables)
  • Determining if Vector Fields are Conservative (Three Variables)
  • Finding a Potential Function (Three Variables)
  • Finding the Divergence of a Vector Field at a Point
    
    
• Line Integrals
  • Evaluating a Line Integral (ds) over a line segment
  • Finding the Mass of a Wire Defined as a Piecewise Curve
  • Finding the Work Done by a Particle Against a Force Field Around a Circle
  • Using the Graph to Determine if Work is Positive, Negative, or Zero
    
    
• The Fundamental Theorem of Line Integrals and Independence of Path
  • Applying the Fundamental Theorem of Line Integrals
  • Finding a Line Integral Over a Conservative Vector Field Using Independence of Path
  • Applying the Fundamental Theorem of Line Integrals (3D)
  • Finding the Work Done by a Conservative Force Field on a Particle Moving Along a Closed Curve
    
    
• Green's Theorem
  • Using Green's Theorem
  • Using Green's Theorem in Polar Coordinates
  • Splitting a Non-Simple Curve Into Two Simple Curves
  • Using Green's Theorem to Find the Area of an Ellipse
    
    
• Parametric Surfaces
  • Eliminating the Parameter
  • Eliminating the Parameter (for a Cylinder)
  • Finding a Vector Valued Function Given a Rectangular Equation that is a Function z = f(x,y)
  • Finding a Vector Valued Function for an Ellipsoid
  • Finding Parametric Equations for a Surface of Revolution about the y-axis
  • Finding a Tangent Plane Given a Parameterized Surface
  • Finding the Surface Area of a Parameterized Surface
    
    
• Surface Integrals
  • Finding the Mass of a Surface Defined by a Function of x and y
  • Finding the Surface Integral Over a Parameterized Surface
  • Finding the Flux Through a Surface
    
    
• The Divergence Theorem
  • Applying the Divergence Theorem
  • Showing that the Surface Integral of CurlF ^. N dS Equals 0
    
    
• Stoke's Theorem
  • Using Stoke's Theorem to Evaluate a Line Integral Over a Triangle
  • Verifying Stoke's Theorem by Computing Both the Line Integral and the Surface Integral

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