Introduction to Functions of Several Variables

Introduction to Functions of Two Variables with Applications

Introduction to Functions of Two Variables

Ex: Function Values of a Function of Two Variables Using a Table

Evaluate a Function of Two Variables (Constants)

Evaluate a Function of Two Variables (Variables)

Determine and Interpret Slopes of a Linear Multivariable Function

Application: Write a Linear Multivariable Function

Ex: Evaluate a Function of Two Variables (Cobb-Douglas Production Function)

Ex: Function Values of a Function of Two Variables (Square Root)

Ex: Function Values of a Function of Two Variables (Polynomial)

Ex: Function Values of a Function of Two Variables (Fraction)

Ex: Function Values of a Function of Two Variables (Exponential)

Ex 1: Determine the Domain of a Function of Two Variables

Ex 2: Determine the Domain of a Function of Two Variables

Level Curves of Function of Two Variables

Ex 1: Determine a Function Value Using a Contour Map

Ex 2: Determine a Function Value Using a Contour Map

Ex: Determine if a Function is Increasing or Decreasing in a Direction Using a Contour Map

Use Traces of a Surface to Select the Equation of the Surface (Ex1)

Use Traces of a Surface to Select the Equation of the Surface (Ex 2)

Determine the Traces of Surface Given the Graph of the Surface

Describe Traces of Quadric Surfaces: y = 1 Trace of an Ellipsoid

Describe Traces of Quadric Surfaces: x = 1 Trace of an Ellipsoid

Describe Traces of Quadric Surfaces: x = 1 Trace of an Paraboloid

Graph a Function of Two Variable Using 3D Calc Plotter

Graph a Contour Plots (Level Curves) Using 3D Calc Plotter

Limits of Functions of Two Variables

Limits of Functions of Two Variables

Find the Limit of a Function of Two Variables: Direct Substitution

Find the Limit of a Function of Two Variables: Rationalize or Factor 1

Find the Limit of a Function of Two Variables: Rationalize or Factor 2

Find the Limit of a Function of Two Variables: Factor Difference of Squares

Find the Limit of a Function of Two Variables: Factor Difference of Cubes

Ex: Limit of a Function of Two Variables (Origin - DNE)

Ex: Limit of a Function of Two Variables (Origin - Exist)

Ex: Limit of a Function of Two Variables (Not Origin - Exist - Direct Substitution)

Ex: Limit of a Function of Two Variables (Not Origin - DNE)

Partial Derivatives of functions of Two or More Variables

First Order Partial Derivatives

Ex: Estimate the Value of a Partial Derivative Using a Contour Map

Ex: Estimate the Value of a Partial Derivative Using a Contour Map

Determining Basic Partial Derivatives: Functions of Two Variables

Determining Partial Derivatives: Functions of Two Variables

Determine Partial Derivatives Using the Chain Rule: Functions of Two Variables

Determining Partial Derivatives: Functions of Three Variables

Ex: Determine a Partial Derivative Function of an Exponential Function of Two Variables

Ex: Determine a Partial Derivative Function of an Polynomial Function of Two Variables

Ex: Find Partial Derivative Values for a Polynomial Function of Two Variables

Ex: Find Partial Derivative Values for a Square Root Function of Two Variables

Determine Partial Derivatives Using the Chain Rule: Square Root Function

Ex: Find the Partial Derivatives of the Cobb Douglas Production Function

Ex: Find the Partial Derivative of a Function of Three Variables (Square Root)

Ex: Application of First Order Partial Derivative (Change in Production)

Second Order Partial Derivatives

Find Second Order Partial Derivatives: f_xy and f_yy

Ex: Find First and Second Order Partial Derivatives

Ex: Determine Second Order Partial Derivatives

Differentials of Functions of Two Variables

Find dz for a Function of Two Variables and Compare to Delta z

Find dz for a Function of Two Variables and Compare to Delta z

Find dz for a Function of Two Variables and Compare to Delta z

Applications of Differentials of Functions of Several Variables

Linear Approximation for a Function Value (Function of Two Variables)

The Chain Rule and Directional Derivatives, and the Gradient Functions of Two Variables

The Chain Rule for Functions of Two Variables With One Independent Variable

Use the Chain Rule to Find a Derivative of a Function of Two Variables and One Independent Variable (Basic)

Use the Chain Rule to Find a Derivative of a Function of Two Variables and One Independent Variable

Use the Chain Rule to Find a Derivative of a Function of Two Variables and Two Independent Variables

The Chain Rule for Functions of Two Variables With Two Independent Variable

Ex: Chain Rule - Function of Two Variables with One Independent Variable

Ex: Chain Rule - Function of Two Variables with Two Independent Variable

Ex: Chain Rule - Function of Two Variables with Three Independent Variable

Application of Chain Rule of a Function of Two Variables - Change of Volume

Implicit Differentiation of Functions in One Variable using Partial Derivatives

Perform Implicit Differentiation Using Partial Derivatives

Partial Implicit Differentiation

Find Partial Derivatives of an Implicit Equation Using Partial Derivatives

Partial Derivatives of Implicit Equation Using Partial Derivatives

Directional Derivatives

Show Tangent Lines to a Surface Using 3D Calc Plotter - Directional Derivative

The Gradient

Show the Gradient to a Surface Using 3D Calc Plotter

Ex: Find the Gradient of the Function f(x,y)=xy

Ex: Find the Gradient of the Function f(x,y)=e^(2x)sin(3y)

Ex: Find the Gradient of the Function f(x,y)=5xsin(xy)

Find the Gradient Vector Field of f(x,y)=x^3y^5

Find the Gradient Vector Field of f(x,y)=ln(2x+5y)

Ex: Use the Gradient to Find the Maximum Rate of Increase of f(x,y)=(4y^5)/x from a Point

Ex 1: Find a Value of a Directional Derivative - f(x,y)=xy

Ex 2: Find a Value of a Directional Derivative - f(x,y)=x^n*y^m

Ex 3: Find a Value of a Directional Derivative - f(x,y)=ln(x^2+y^2)

Normal Vectors and Tangent Planes to Functions of Two Variables

The Equation of a Tangent Plane to a Surface (Relating to Tangent Line)

Derive or Prove the Equation of a Tangent Line to a Surface

Find the Equation of the Tangent Plane to a Surface - f(x,y)=-2x^2+4y^2-4y

Find the Equation of the Tangent Plane to a Surface - f(x,y)=2e^(x^2-2y)

Find the Equation of the Tangent Plane to a Surface - f(x,y)=2ycos(5x-3y)

Determining a Unit Normal Vector to a Surface

Verifying the Equation of a Tangent Plane to a Surface

Determining the Equation of a Tangent Plane

Ex 1: Find the Equation of a Tangent Plane to a Surface

Ex 2: Find the Equation of a Tangent Plane to a Surface (Exponential)

Ex 3: Find the Equation of a Tangent Plane to a Surface (Trigonometric)

Find a Linear Approximation to a Function of Two Variables and Estimate a Function Value

Relative Extrema and Applications to Functions of Two Variables

Determine the Critical Points of a Function of Two Variables (Basic)

Find the Critical Point of a Function of Two Variables

Determine the Critical Points of a Function of Two Variables

Determining the Relative Extrema of a Function of Two Variables

Determine Relative Extrema of a Function of Two Variables: Basic #1

Determine Relative Extrema of a Function of Two Variables: Basic #2

Determine Relative Extrema of a Function of Two Variables

Find Relative Extrema and Saddle Point of a Function of Two Variables

Extrema of Functions of Two Variables (Unconstrained) f=x^4+16xy^2+2y^4

Extrema of Functions of Two Variables (Unconstrained) f=x^6+y^6+6/(xy)

Ex 1: Classify Critical Points as Extrema or Saddle Points - Function of Two Variables

Ex 2: Classify Critical Points as Extrema or Saddle Points - Function of Two Variables

Ex: Determine Relative Extrema for a Function of Two Variables

Absolute Extrema of Functions of Two Variables

Absolute Extrema of a Function of Two Variable Over Bounded Region (Basic)

Find the Absolute Extrema of a Function of Two Variable Over a Triangular Region

Absolute Extrema of a Function of Two Variable Over Bounded Region (Rectangle)

Absolute Extrema of a Function of Two Variable Over Bounded Region (Circle)

Applications of Extrema of Functions of Two Variables I

Applications of Extrema of Functions of Two Variables II

Applications of Extrema of Functions of Two Variables III

Ex: Determine the Quantity to Maximize Revenue - Function of Two Variables

Ex: Minimize Cost to Make Open Top Box - Function of Two Variables

Application of Relative Extrema of a Function of Two Variables (Brand Loyalty)

Find the Minimum Distance Between a Cone and a Point

Lagrange Multipliers

Lagrange Multipliers - Part 1

Lagrange Multipliers - Part 2

Lagrange Multipliers: Maximized f=xy Under Constraint x+5y=64

Lagrange Multipliers: Relative Extrema of f=xy Under Constraint y=9-x^2

Lagrange Multipliers: Minimize f=x^2+y^2 under Constraint x+4y=20

Lagrange Multipliers: Abs Extrema of Paraboloid Under Circular Constraint

Absolute Extrema of a Function of Two Variable Over Bounded Region (Circle / Lagrange)

Maximize a Cobb Douglas Production Function Using Lagrange Multipliers

Maximize a Function of Two Variable Under a Constraint Using Lagrange Multipliers - f(x,y)=x^2y

Lagrange Multipliers App: Maximize the Area of a Rectangle

Lagrange Multipliers App: Minimize the Cost of a Box

Minimize a Cost Function of Two Variable Under a Constraint Using Lagrange Multipliers

Lagrange Multipliers: Find the Max and Min of a Function of Two Variables

Lagrange Multipliers: Find Minimum Cost using Cobb Douglas Production Function

Minimize the Total Energy Generated through a Circuit Using Lagrange Multipliers

Approximate the Volume of Pool With The Midpoint Rule Using a Table of Values

Double Integral Approximation Using Midpoint Rule Using Level Curves

Ex: Double Integral Approximation Using Midpoint Rule - f(x,y)=ax+by

Integrating Functions of Two Variables

Introduction to Double Integrals and Volume

Double Integrals and Volume over a General Region - Part 1

Double Integrals and Volume over a General Region - Part 2

Evaluating Double Integrals

Ex: Double Integrals - Describe a Region of Integration (Triangle)

Ex: Double Integrals - Describe a Region of Integration (Quadratic)

Ex: Double Integrals - Describe a Region of Integration (Advanced)

Ex 1: Evaluate a Double Integral Over a Rectangular Region to Find a Volume - f(x,y)=c

Ex 2: Evaluate a Double Integral Over a Rectangular Region to Find a Volume - f(x,y)=c

Ex: Evaluate a Double Integral Over a Rectangular Region to Find a Volume - f(x,y)=ax

Ex: Evaluate a Double Integral Over a Rectangular Region - f(x,y)=ax+by

Ex: Evaluate a Double Integral to Determine Volume (Basic)

Ex: Evaluate a Double Integral to Determine Volume - Change Order of Integration

Ex: Evaluate a Double Integral Over a Rectangular Region to Find a Volume - f(x,y)=x/y

Use a Double Integral to Find the Volume Under a Paraboloid Over a Rectangular Region

Find Volume Using a Double Integral Over Triangular Region (Basic)

Find Volume Using a Double Integral Over Triangular Region

Evaluate a Double Integral Using Substitution Over a Rectangular Region - f(x,y)=(xy^2)/(x^2+1)

Evaluate a Double Integral Using Substitution Over a Rectangular Region - f(x,y)=(ax+by)^n

Evaluate a Double Integral Using Substitution Over a Rectangular Region - f(x,y)=xysin(x^2+y^2)

Evaluate a Double Integral over a Trapezoidal Region Requiring Integration by Part

Evaluate a Double Integral Over a General Region - f(x,y)=ax+by

Evaluate a Double Integral Over a General Region - f(x,y)=xy^2

Evaluate a Double Integral Over a General Region with Substitution - f(x,y)=e^(x/y)

Evaluate Double Integral of y/(1+x) over Region Bounded by y=-x and y=sqrt(x)

Average Value of a Function of Two Variables

Fubini's Theorem

Setting up a Double Integral Using Both Orders of Integration

Double Integrals: Changing the Order of Integration

Double Integrals: Changing the Order of Integration - Example 1

Double Integrals: Changing the Order of Integration - Example 2

Double Integrals in Polar Coordinates

Introduction to Double Integrals in Polar Coordinates

Write a Double Integral Using Polar Coordinates and Evaluate (1)

Write a Double Integral Using Polar Coordinates and Evaluate (2)

Double Integrals in Polar Coordinates - Example 1

Double Integrals in Polar Coordinates - Example 2

Area Using Double Integrals in Polar Coordinates - Example 1

Area Using Double Integrals in Polar Coordinates - Example 2

Double Integrals in Polar Form - Volume of a Right Circular Cylinder (f(x,y) over a circle)

Double Integrals in Polar Form - Volume of a Half Sphere Over a Circle

Evaluate a Double Integral in Polar Form - f(x,y)=ax+by Over a Half-Circle

Evaluate a Double Integral in Polar Form - f(x,y)=cos(x^2+y^2) Over a Ring

Volume of a Drilled Sphere Using a Double Integral in Polar Form

Double Integrals in Polar Form - Volume Bounded by Two Paraboloids

Applications of Double Integrals: Mass, Center of Mass, Jacobian

Double Integral Application: Mass of a Triangular Region on the Coordinate Plane

Double Integral Application: Mass of a Circular Region (Polar Coordinates)

Double Integrals - Find the Mass of a Lamina Over a Region in the xy Plane

Double Integrals - Find the Center Mass of a Lamina Over a Region Using Polar Coordinates

Double Integral Application: Mass, Moments, Center of Mass of Triangular Region

Double Integrals - Find the Total Charge Over a Triangular Region

Double Integrals - Find a Probability Using the Exponential Density Function: P(x<a,y<b)

Double Integrals - Surface Area over a Rectangular Region (Basic)

Double Integrals - Surface Area over a Circle Using Polar Coordinates (Basic)

Surface Area of a Cone Bounded by Two Planes Using a Double Integral (Polar)

Find the Jacobian Given x=au+bv, y=u^2+cv

Determine a Jacobian (2 by 2)

Evaluate a Double Integral of ax+by Over Parallelogram Given Transformation Equations (Jacobian)

Evaluate a Double Integral of ax+by Over Parallelogram (Jacobian)

Evaluate a Double Integral of x^2 Over an Ellipse Using a Change of Variables (Jacobian, Polar)

Triple Integrals

Introduction to Triple Integrals

Evaluating Triple Integrals – Example

Ex 1: Set Up and Evaluate a Triple Integral of z - Part 1: Limits of Integration

Ex 1: Set Up and Evaluate a Triple Integral of z - Part 2: Evaluate the Triple Integral

Ex 2: Set up and Evaluate a Triple Integral of 2xz

Ex 3: Set Up and Evaluate a Triple Integral of y - Part 1: Limits of Integration

Ex 3: Set Up and Evaluate a Triple Integral of y - Part 2: Evaluate the Triple Integral

Ex 4: Set up and Evaluate a Triple Integral of x+y-4z

Evaluate a Triple Integral Using a Change of Variables (Jacobian)

Triple Integrals and Volume - Part 1

Triple Integrals and Volume - Part 2

Triple Integrals and Volume - Part 3

Set up a Triple Integral to Determine Volume (Rectangular Coordinates)

Use a Triple Integral to Find the Volume of a Spherical Cap

Determine Limits of Integration for a Triple Integral - Region of Integration is a Tetrahedron

Use a Triple Integral to Determine Volume Ex 1 (Rectangular Coordinates)

Use a Triple Integral to Determine Volume Ex 2 (Rectangular Coordinates)

Triple Integrals: Find the Volume of a Tetrahedron Given the Vertices

Application of Triple Integrals: Mass

Changing the Order of Triple Integrals

Triple Integrals in Cylindrical and Spherical Coordinates

Write a Cylindrical Equations in Rectangular Form

Write a Rectangular Equations in Cylindrical Form

Write Spherical Equations in Rectangular Form

Convert a Spherical Equation to a Rectangular Equation rho=2csc(phi)

Convert a Spherical Equation to a Rectangular Equation phi=pi/6

Convert a Rectangular Equation to a Spherical Equation x^2+y^2+z^2-9z=0

Convert a Rectangular Equation to a Spherical Equation x^2+y^2-z^2=0

Triple Integrals Using Cylindrical Coordinates

Triple Integral and Volume Using Cylindrical Coordinates

Rewrite Triple Integrals Using Cylindrical Coordinates

Use a Triple Integral to Determine Volume Ex 1 (Cylindrical Coordinates)

Use a Triple Integral to Find the Volume Bounded by Two Paraboloid (Cylindrical)

Introduction to Triple Integrals Using Spherical Coordinates

Triple Integrals and Volume using Spherical Coordinates

Evaluate a Triple Integral Using Cylindrical Coordinates - Triple Integral of e^z

Evaluate a Triple Integral Using Spherical Coordinates - Triple Integral of 1/(x^2+y^2+z^2)

Find the Moment of Inertia about the z-axis of a Solid Using Triple Integrals

Find the Center of Mass of a Solid Using Triple Integrals

Use a Triple Integral to Determine the Mass Bounded by Two Paraboloids (Cyl)

Use a Triple Integral to Determine the Mass of an Cone Cut From Sphere (Spherical)

A Change of Variables for a Double Integral: Jacobian

Example of a Change of Variables for a Double Integral: Jacobian

A Change of Variables for a Triple Integral: Jacobian

Vector Fields, Divergence, and Curl

Introduction to Vector Fields

Graph 2D Vector Fields Using CalcPlot3D

The Divergence of a Vector Field

Determine the Divergence of a 2D Vector Field (Basic) Determine the Divergence of a 2D Vector Field (Trig)

Ex 1: Determine the Divergence of a Vector Field

Ex 2: Determine the Divergence of a Vector Field

Ex: Determine the Sign of the Divergence from the Graph of a Vector Field

The Curl of a Vector Field

Ex 1: Determine the Curl of a Vector Field

Ex 2: Determine the Curl of a Vector Field

Ex 1: Determine the Curl of a Vector Field (2D)

Ex 2: Determine the Curl of a Vector Field (2D)

Conservative Vector Fields

Line Integrals

Defining a Smooth Parameterization of a Path

Ex 1A: Determine a Piecewise Smooth Parameterization for a Curve (Triangle)

Ex 1B: Determine a Piecewise Smooth Parameterization for a Curve (Triangle)

Ex 2: Determine a Piecewise Smooth Parameterization for a Curve

Ex: Determine Parametric Equations for an Ellipse

Line Integrals in R^2

Line Integrals in R^3

Line Integral of Vector Fields

Line Integrals in Differential Form

Evaluate a Line Integral of y with Respect to Arc Length (Area)

Evaluate a Line Integral of xy^2 with Respect to Arc Length C: Half Circle (Area)

Evaluate a Line Integral of xy with Respect to Arc Length (Mass of Wire)

Evaluate a Line Integral of x^2+y^2+z^2 with Respect to Arc Length(Mass of Wire)

Evaluate a Line Integral of x^3y^2 with Respect to x (Differential Form)

Evaluate the Line Integral of x^2z Along a Line Segment in 3D

Evaluate a Line Integral of in Differential Form

Evaluate a Line Integral of F*dr

Evaluate a Line Integral of F*dr Around a Circle

Line Integral Application - Work of a Charged Particle

Line Integral Application: Determine the Mass of a Wire

Determining the Potential Function of a Conservative Vector Field

The Fundamental Theorem of Line Integrals - Part 1

The Fundamental Theorem of Line Integrals - Part 2

Fundamental Theorem of Line Integrals - Closed Path/Curve

Ex 1: Fundamental Theorem of Line Integrals - Given Vector Field in a Plane

Ex 2: Fundamental Theorem of Line Integrals - Given Vector Field in a Plane (Not Conservative)

Ex 3: Fundamental Theorem of Line Integrals - Given Vector Field in a Plane

Ex 4: Fundamental Theorem of Line Integrals - Given Vector Field in Space

Green's Theorem - Part 1

Green's Theorem - Part 2

Evaluate a Line Integral using Green's Theorem

Ex: Use Green's Theorem to Evaluate a Line Integral (Rectangle)

Evaluate a Line Integral of F*dr Around a Circle with Green's Theorem

Ex: Use Green's Theorem to Evaluate a Line Integral (Polar)

Ex: Use Green's Theorem to Evaluate a Line Integral (Negative Orientation)

Ex: Use Green's Theorem to Determine Area of a Region Enclosed by a Curve

Determining Area using Line Integrals

Flux Form of Green's Theorem

Determine the Flux of a 2D Vector Field Using Green's Theorem (Rectangle)

Determine the Flux of a 2D Vector Field Using Green's Theorem (Parabola)

Determine the Flux of a 2D Vector Field Using Green's Theorem (Hole)

Surface Integrals

Parameterized Surfaces

Write a Parameterized Surface Using Cartesian (Rectangular) Coordinates

Graph Parameterized Surfaces using 3D Calc Plotter

Area of a Parameterized Surface

Surface Integral with Explicit Surface Part 1

Surface Integral with Explicit Surface Part 2

Ex: Surface Area of a Function of Two Variables (Surface Integral)

Surface Integrals with Parameterized Surface - Part 1

Surface Integrals with Parameterized Surface - Part 2

Double Integrals - Surface Area of a Vector Valued Function Over a Region

Ex: Surface Area of a Parametric Surface (Surface Integral)

Surface Integral of a Vector Field - Part 1

Surface Integral of a Vector Field - Part 2

Ex: Evaluate a Surface Integral (Parametric Surface - Helicoid)

Ex: Evaluate a Surface Integral (Basic Explicit Surface - Plane Over Rectangle)

Ex: Evaluate a Surface Integral Using Polar Coordinates- Implicit Surface (Cone)

Ex: Evaluate a Flux Integral with Surface Given Explicitly (rectangle)

Ex: Evaluate a Flux Integral with Surface Given Parametrically (semicircle)

Ex: Using a Flux Integral to Determine a Mass Flow Rate

Stoke's Theorem - Part 1

Stoke's Theorem - Part 2

Ex 1: Using Stoke's Theorem to Evaluate a Line Integral as a Surface Integral

Ex 2: Using Stoke's Theorem to Evaluate a Line Integral as a Surface Integral

Ex 1: Using Stoke's Theorem to Evaluate a Surface Integral as a Line Integral

Ex 2: Using Stoke's Theorem to Evaluate a Surface Integral as a Line Integral

The Divergence Theorem - Part 1

The Divergence Theorem - Part 2

Ex: Use the Divergence Theorem to Evaluate a Flux Integral (Rectangular Coordinates)

Ex: Use the Divergence Theorem to Evaluate a Flux Integral (Cylindrical Coordinates)

Ex: Use the Divergence Theorem to Evaluate a Flux Integral (Spherical Coordinates)

Use the Divergence Theorem to Determine the Flux Across a Hemisphere

Use the Divergence Theorem to Determine the Flux a Tetrahedron

Graphing Calculator

Determine the value of the derivative function on the graphing calculator

Determining the value of a definite integral on the graphing calculator

Sequences on the TI84 Graphing Calculator

Sequences and Series on the TI84

Graph Partial Sums of an Infinite Series on the TI84

Graphing Parametric Equations in the TI84

3D Calc Plotter Videos

Graph 2D Vector Fields Using CalcPlot3D

Graph Points in 3D Using 3D Calc Plotter

Graph 3D Vectors Using 3D Calc Plotter

Graph Tangent Planes to Surfaces Using 3D Calc Plotter

Graph a Function of Two Variable Using 3D Calc Plotter

Graph a Contour Plots (Level Curves) Using 3D Calc Plotter

Graph Parameterized Surfaces using 3D Calc Plotter

Graph Implicit Equations (Quadric Surfaces) Using 3D Calc Plotter

Graph Space Curves Given as a Vector Function Using 3D Calc Plotter

Graph Tangent Lines to a Surface in the x and y Direction (Partial Derivatives)