Functions of Several Variables
29 min 6 Examples
- Overview of Functions of Several Variables
- Example of Evaluating and finding the domain of a function of several variables
- Example #2 Find and Sketch domain of a function of several variables
- Example #3 Find and Sketch the domain of a function of several variables
- Example #4 Find the domain and Range of a function of several variables
- Example #1 of Sketching a 3D function and finding its domain and range
- Example #2 of Sketching a 3D function and finding its domain and range
Multivariable Limits
1 hr 2 min 2 Examples
- Overview of Continuity and Discontinuity
- Example #1 for proving a function is continuous or discontinuous
- Example #2 for proving a function is continuous or discontinuous
Partial Derivative
58 min 9 Examples
- Partial Differentiation Overview
- Example of how a function increases/decreases using partial derivatives
- Example #1 of Finding First Order Partial Derivatives
- Example #2 of Finding First Order Partial Derivatives
- Example #3 of Finding First Order Partial Derivatives
- Example #1 of finding slope of the tangent when a surface intersects a plane
- Example #2 of finding slope of the tangent when a surface intersects a plane
- Example of using partial derivatives to find the rate of change of volume
- Overview of Higher Order Partial Derivatives
- Example #1 of Finding Second Order Derivatives
- Example #2 of Finding Second Order Partial Derivatives
- Example of finding a Higher-Order Partial Derivative
Tangent Planes & Linear Approximations
1 hr 20 min 20 Examples
- Overview of Tangent Lines and Linear Approximation for Single Variable Calculus
- Overview of Tangent Planes and Linear Approximation for Multivariable Calculus
- Example #1 of finding Linear Approximation using a tangent line
- Example #2 of finding Linear Approximation using a tangent line
- Example #3 of using a linear approximation using a tangent line
- Example #4 of using a linear approximation using a tangent line
- Example #1 of finding Linear Approximation using the tangent plane
- Example #2 of finding Linear Approximation using a tangent plane
- Example of finding where a tangent plane is horizontal
- Overview of Differentiability in Multivariable Calculus
- Example of showing a function of two or more variables is differentiable
- Definition and Example for Finding the Total Differential of a Function
- Example of using the Total Differential in finding max error for a circular spa
- Example of using the Total Differential in finding max error for a rectangular box
- Example of using the Total Differential in estimating the amount of tin for a can
Multivariable Chain Rule
1 hr 6 min 10 Examples
- Overview of the Chain Rule for Single Variable Calculus
- Two Examples of using the Chain Rule for 2D
- Overview of the Two Cases of the Chain Rule for Multivariable Calculus
- Example #1 of Chain Rule for 3D – Case 1
- Example #2 of Chain Rule for 3D – Case 1
- Example #3 of Chain Rule for 3D – Case 1
- Example #4 of Chain Rule for 3D – Case 2
- Example #5 of Chain Rule for 3D – Case 2
- Overview of the General Version of the Chain Rule
- Example of General Version of Chain Rule for 3D
- Overview of Implicit Differentiation for 2D and 3D
- Proof of Implicit Differentiation for 3D
- Example #1 of Implicit Differentiation for 3D
- Example #2 of Implicit Differentiation for 3D
Directional Derivatives & Gradient Vectors
58 min 7 Examples
- What is a Directional Derivative?
- Definition of the Directional Derivative Analytically and Geometrically
- Example #1 of finding the Directional Derivative given a vector
- Example #2 of finding the Directional Derivative given a vector
- Example #3 of finding the Directional Derivative given an angle
- Importance of the Gradient Vector for finding maximum increase
- Example #1 of finding the maximum rate of change and its direction
- Example #2 of finding the maximum rate of change and its direction
- Overview of Tangent Planes to Level Surfaces
- Example #1 of writing an equation of a tangent plane to a surface
- Example #2 of writing an equation of a tangent plan to a surface in symmetric form, and finding a normal line
- Recap of the Significance of the Gradient Vector
Saddle Point
52 min 5 Examples
- How did we find relative maximums and minimums in Calc 1
- Example of finding extrema for single variable calculus
- How do we find relative maximums and minimums in Calc 3?
- Example #1 of max/min values for a function of several variables
- Example #2 of max/min values for a function of several variables
- Example #3 of max/min values for a function of several variables
- Overview of Optimization for Multivariable Calculus
- Example of optimization for a function of several variables
Extrema Of Multivariable Functions
33 min 3 Examples
- How did we find Absolute Maximums and Minimums in Calc 1?
- Example of finding Absolute extrema for single variable calculus
- How do we find Absolute Maximums and Minimums in Calc 3?
- Example of finding Absolute Extrema on a disk
- Example of finding Absolute Extrema on a rectangle
Lagrange Multiplier
41 min 3 Examples
- Overview of how and why we use Lagrange Multipliers to find Absolute Extrema
- Steps for how to optimize a function using Lagrange multipliers
- Example #1 of using Lagrange multipliers given one constraint
- Example #2 of using Lagrange multipliers given two constraints
- Example #3 of using Lagrange multipliers given an inequality
Chapter Test
2 hr 35 min 19 Practice Problems
- Determine whether the statement is true or false (Problems #1-3)
- Determine the domain of the functions and sketch the domain (Problem #4)
- Evaluate the limit or show that it does not exist (Problems #5-6)
- Find the first partial derivatives for each function (Problems #7-8)
- Find all second order partial derivatives for the given function (Problem #9)
- Find an equation of a tangent line to the surface at a point (Problem #10)
- Find the partial derivatives implicitly (Problem #11)
- Find the directional derivative (Problem #12)
- Find the maximum rate of change and in what direction does it occur? (Problem #13)
- Find all local maximums, minimum or saddle points (Problem #14)
- Find all local maximums, minimum or saddle points (Problem #15)
- Find the absolute maximum and minimum values of the function (Problem #16)
- Find the dimensions of a package with largest volume (Problem #17)
- Use Lagrange multipliers to find the maximum and minimum values (Problem #18)
- Use Lagrange multipliers to find the maximum and minimum values (Problem #19)
