Multivariable Calculus Tutorials from Khan Academy

Video Lectures

Displaying all 86 video lectures.
Lecture 1
Multivariable functions
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Multivariable functions
An introduction to multivariable functions, and a welcome to the multivariable calculus content as a whole.
Lecture 2
Representing points in 3d
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Representing points in 3d
Learn how to represent and think about points and vectors in three-dimensional space.
Lecture 3
Introduction to 3d graphs
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Introduction to 3d graphs
Three-dimensional graphs are a way to represent functions with a two-dimensional input and a one-dimensional output.
Lecture 4
Interpreting graphs with slices
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Interpreting graphs with slices
3d graphs can be a lot to take in, but it helps to imagine slicing them with planes parallel to the x-axis or y-axis and relate them with two-dimensional graphs.
Lecture 5
Contour plots
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Contour plots
An alternative method to representing multivariable functions with a two-dimensional input and a one-dimensional output, contour maps involve drawing purely in the input space.
Lecture 6
Parametric curves
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Parametric curves
When a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space.
Lecture 7
Parametric surfaces
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Parametric surfaces
Functions that have a two-dimensional input and a three-dimensional output can be thought of as drawing a surface in three-dimensional space. This is actually pretty cool.
Lecture 8
Vector fields, introduction
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Vector fields, introduction
Vector fields let you visualize a function with a two-dimensional input and a two-dimensional output. You end up with, well, a field of vectors sitting at various points in two-dimensional space.
Lecture 9
Fluid flow and vector fields
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Fluid flow and vector fields
A neat way to interpret a vector field is to imagine that it represents some kind of fluid flow.
Lecture 10
3d vector fields, introduction
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3d vector fields, introduction
Vector fields can also be three-dimensional, though this can be a bit trickier to visualize.
Lecture 11
3d vector field example
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3d vector field example
See an example of how you can start to understand how the formula for a three-dimensional vector field relates to the way it looks.
Lecture 12
Transformations, part 1
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Transformations, part 1
One fun way to think about functions is to imagine that they literally move the points from the input space over to the output space. See what this looks like with some one-dimensional examples.
Lecture 13
Transformations, part 2
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Transformations, part 2
More transformations, but this time with a function that maps two dimensions to two dimensions.
Lecture 14
Transformations, part 3
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Transformations, part 3
Learn how you can think about a parametric surface as a certain kind of transformation.
Lecture 15
Partial derivatives, introduction
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Partial derivatives, introduction
Partial derivatives tell you how a multivariable function changes as you tweak just one of the variables in its input.
Lecture 16
Partial derivatives and graphs
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Partial derivatives and graphs
One of the best ways to think about partial derivatives is by slicing the graph of a multivariable function.
Lecture 17
Formal definition of partial derivatives
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Formal definition of partial derivatives
Partial derivatives are formally defined using a limit, much like ordinary derivatives.
Lecture 18
Symmetry of second partial derivatives
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Symmetry of second partial derivatives
There are many ways to take a "second partial derivative", but some of them secretly turn out to be the same thing.
Lecture 19
Gradient
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Gradient
The gradient captures all the partial derivative information of a scalar-valued multivariable function.
Lecture 20
Gradient and graphs
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Gradient and graphs
Learn how the gradient can be thought of as pointing in the "direction of steepest ascent". This is a rather important interpretation for the gradient.
Lecture 21
Directional derivative
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Directional derivative
Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space.
Lecture 22
Directional derivative, formal definition
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Directional derivative, formal definition
Learn the limit definition of a directional derivative. This helps to clarify what it is really doing.
Lecture 23
Directional derivatives and slope
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Directional derivatives and slope
The directional derivative can be used to compute the slope of a slice of a graph, but you must be careful to use a unit vector.
Lecture 24
Why the gradient is the direction of steepest ascent
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Why the gradient is the direction of steepest ascent
The way we compute the gradient seems unrelated to its interpretation as the direction of steepest ascent. Here you can see how the two relate.
Lecture 25
Gradient and contour maps
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Gradient and contour maps
Gradient vectors always point perpendicular to contour lines.
Lecture 26
Position vector valued functions | Multivariable Calculus | Khan Academy
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Position vector valued functions | Multivariable Calculus | Khan Academy
Using a position vector valued function to describe a curve or path

Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line...

Missed the previous lesson?
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Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.





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Lecture 27
Derivative of a position vector valued function | Multivariable Calculus | Khan Academy
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Derivative of a position vector valued function | Multivariable Calculus | Khan Academy
Visualizing the derivative of a position vector valued function

Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line...

Missed the previous lesson?
https://www.khanacademy.org/math/multivariable-calculus/line...

Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.





Subscribe to KhanAcademy’s Multivariable Calculus channel:
https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub...
Lecture 28
Differential of a vector valued function | Multivariable Calculus | Khan Academy
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Differential of a vector valued function | Multivariable Calculus | Khan Academy
Understanding the differential of a vector valued function

Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line...

Missed the previous lesson?
https://www.khanacademy.org/math/multivariable-calculus/line...

Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.





Subscribe to KhanAcademy’s Multivariable Calculus channel:
https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub...
Lecture 29
Vector valued function derivative example | Multivariable Calculus | Khan Academy
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Vector valued function derivative example | Multivariable Calculus | Khan Academy
Concrete example of the derivative of a vector valued function to better understand what it means

Watch the next lesson: https://www.khanacademy.org/math/multivariable-calculus/line...

Missed the previous lesson?
https://www.khanacademy.org/math/multivariable-calculus/line...

Multivariable Calculus on Khan Academy: Think calculus. Then think algebra II and working with two variables in a single equation. Now generalize and combine these two mathematical concepts, and you begin to see some of what Multivariable calculus entails, only now include multi dimensional thinking. Typical concepts or operations may include: limits and continuity, partial differentiation, multiple integration, scalar functions, and fundamental theorem of calculus in multiple dimensions.





Subscribe to KhanAcademy’s Multivariable Calculus channel:
https://www.youtube.com/channel/UCQQZDc22yCogOyx6DwXt-Ig?sub...
Lecture 30
Multivariable chain rule
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Multivariable chain rule
This is the simplest case of taking the derivative of a composition involving multivariable functions.
Lecture 31
Multivariable chain rule intuition
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Multivariable chain rule intuition
Get a feel for what the multivariable is really saying, and how thinking about various "nudges" in space makes it intuitive.
Lecture 32
Vector form of the multivariable chain rule
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Vector form of the multivariable chain rule
The multivariable chain rule is more often expressed in terms of the gradient and a vector-valued derivative. This makes it look very analogous to the single-variable chain rule.
Lecture 33
Multivariable chain rule and directional derivatives
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Multivariable chain rule and directional derivatives
See how the multivariable chain rule can be expressed in terms of the directional derivative.
Lecture 34
More formal treatment of multivariable chain rule
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More formal treatment of multivariable chain rule
For those of you who want to see how the multivariable chain rule looks in the context of the limit definitions of various forms of the derivative.
Lecture 35
Curvature intuition
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Curvature intuition
An introduction to curvature, the radius of curvature, and how you can think about each one geometrically.
Lecture 36
Curvature formula, part 1
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Curvature formula, part 1
Curvature is computed by first finding a unit tangent vector function, then finding its derivative with respect to arc length. Here we start thinking about what that means.
Lecture 37
Curvature formula, part 2
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Curvature formula, part 2
A continuation in explaining how curvature is computed, with the formula for a circle as a guiding example.
Lecture 38
Curvature formula, part 3
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Curvature formula, part 3
Here, this concludes the explanation for how curvature is the derivative of a unit tangent vector with respect to length.
Lecture 39
Curvature formula, part 4
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Curvature formula, part 4
After the last video made reference to an explicit curvature formula, here you can start to get an intuition for why that seemingly unrelated formula describes curvature.
Lecture 40
Curvature formula, part 5
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Curvature formula, part 5
Here, we finish the intuition for how the curvature relates to the cross product between the first two derivatives of a parametric function.
Lecture 41
Curvature of a helix, part 1
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Curvature of a helix, part 1
An example of computing curvature by finding the unit tangent vector function, then computing its derivative with respect to arc length.
Lecture 42
Curvature of a helix, part 2
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Curvature of a helix, part 2
This finishes up the helix-curvature example started in the last video.
Lecture 43
Curvature of a cycloid
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Curvature of a cycloid
An example of computing curvature with the explicit formula.
Lecture 44
Computing the partial derivative of a vector-valued function
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Computing the partial derivative of a vector-valued function
When a function has a multidimensional input, and a multidimensional output, you can take its partial derivative by computing the partial derivative of each component in the output.
Lecture 45
Partial derivative of a parametric surface, part 1
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Partial derivative of a parametric surface, part 1
When a vector-valued function represents a parametric surface, how do you interpret its partial derivative?
Lecture 46
Partial derivative of a parametric surface, part 2
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Partial derivative of a parametric surface, part 2
Taking the same example surface used in the last example, we now take a look at the partial derivative in the other direction.
Lecture 47
Partial derivatives of vector fields
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Partial derivatives of vector fields
How do you intepret the partial derivatives of the function which defines a vector field?
Lecture 48
Partial derivatives of vector fields, component by component
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Partial derivatives of vector fields, component by component
Here we step through each partial derivative of each component in a vector field, and understand what each means geometrically.
Lecture 49
Divergence intuition, part 1
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Divergence intuition, part 1
Vector fields can be thought of as representing fluid flow, and divergence is all about studying the change in fluid density during that flow.
Lecture 50
Divergence intuition, part 2
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Divergence intuition, part 2
In preparation for finding the formula for divergence, we start getting an intuition for what points of positive, negative and zero divergence should look like.
Lecture 51
Divergence formula, part 1
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Divergence formula, part 1
How does the x-component of a vector field relate to the divergence?
Lecture 52
Divergence formula, part 2
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Divergence formula, part 2
Here we finish the line of reasoning which leads to the formula for divergence in two dimensions.
Lecture 53
Divergence example
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Divergence example
An example of computing and interpreting the divergence of a two-dimensional vector field.
Lecture 54
Divergence notation
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Divergence notation
Learn how divergence is expressed using the same upsidedown triangle symbols that the gradient uses.
Lecture 55
2d curl intuition
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2d curl intuition
A description of how vector fields relate to fluid rotation, laying the intuition for what the operation of curl represents.
Lecture 56
2d curl formula
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2d curl formula
Here we build up to the formula for computing the two-dimensional curl of a vector field, reasoning through what partial derivative information corresponds to fluid rotation.
Lecture 57
2d curl example
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2d curl example
A worked example of computing and interpreting two-dimensional curl.
Lecture 58
2d curl nuance
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2d curl nuance
The meaning of positive curl in a fluid flow can sometimes look a bit different from the clear cut rotation-around-a-point examples discussed in previous videos.
Lecture 59
Describing rotation in 3d with a vector
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Describing rotation in 3d with a vector
Learn how a three-dimensional vector can be used to describe three-dimensional rotation. This is important for understanding three-dimensional curl.
Lecture 60
3d curl intuition, part 1
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3d curl intuition, part 1
Here we start transitioning from the understanding of two-dimensional curl into an understanding of three-dimensional curl.
Lecture 61
3d curl intuition, part 2
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3d curl intuition, part 2
Continuing the intuition for how three-dimensional curl represents rotation in three-dimensional fluid flow.
Lecture 62
3d curl formula, part 1
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3d curl formula, part 1
How to compute a three-dimensional curl, imagined as a cross product of sorts.
Lecture 63
3d curl formula, part 2
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3d curl formula, part 2
This finishes the demonstration of how to compute three-dimensional curl using a certain determinant.
Lecture 64
3d curl computation example
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3d curl computation example
A worked example of a three-dimensional curl computation.
Lecture 65
Laplacian intuition
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Laplacian intuition
A visual understanding for how the Laplace operator is an extension of the second derivative to multivariable functions.
Lecture 66
Laplacian computation example
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Laplacian computation example
A worked example of computing the laplacian of a two-variable function.
Lecture 67
Explicit Laplacian formula
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Explicit Laplacian formula
This is another way you might see the Laplace operator written.
Lecture 68
Harmonic Functions
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Harmonic Functions
If the Laplacian of a function is zero everywhere, it is called Harmonic. Harmonic functions arise all the time in physics, capturing a certain notion of "stability", whenever one point in space is influenced by its neighbors.
Lecture 69
What is a tangent plane
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What is a tangent plane
The "tangent plane" of the graph of a function is, well, a two-dimensional plane that is tangent to this graph. Here you can see what that looks like.
Lecture 70
Controlling a plane in space
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Controlling a plane in space
How can you describe a specified plane in space as the graph of a function?
Lecture 71
Computing a tangent plane
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Computing a tangent plane
Here you can see how to use the control over functions whose graphs are planes, as introduced in the last video, to find the tangent plane to a function graph.
Lecture 72
Local linearization
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Local linearization
A "local linearization" is the generalization of tangent plane functions; one that can apply to multivariable functions with any number of inputs.
Lecture 73
What do quadratic approximations look like
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What do quadratic approximations look like
After learning about local linearizations of multivariable functions, the next step is to understand how to approximate a function even more closely with a quadratic approximation.
Lecture 74
Quadratic approximation formula, part 1
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Quadratic approximation formula, part 1
How to creat a quadratic function that approximates an arbitrary two-variable function.
Lecture 75
Quadratic approximation formula, part 2
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Quadratic approximation formula, part 2
A continuation from the previous video, leading to the full formula for the quadratic approximation of a two-variable function.
Lecture 76
Quadratic approximation example
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Quadratic approximation example
A worked example for finding the quadratic approximation of a two-variable function.
Lecture 77
The Hessian matrix
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The Hessian matrix
The Hessian matrix is a way of organizing all the second partial derivative information of a multivariable function.
Lecture 78
Expressing a quadratic form with a matrix
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Expressing a quadratic form with a matrix
How to write an expression like ax^2 + bxy + cy^2 using matrices and vectors.
Lecture 79
Vector form of multivariable quadratic approximation
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Vector form of multivariable quadratic approximation
This is the more general form of a quadratic approximation for a scalar-valued multivariable function. It is analogous to a quadratic Taylor polynomial in the single-variable world.
Lecture 80
Multivariable maxima and minima
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Multivariable maxima and minima
A description of maxima and minima of multivariable functions, what they look like, and a little bit about how to find them.
Lecture 81
Saddle points
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Saddle points
Just because the tangent plane to a multivariable function is flat, it doesn't mean that point is a local minimum or a local maximum. There is a third possibility, new to multivariable calculus, called a "saddle point".
Lecture 82
Warm up to the second partial derivative test
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Warm up to the second partial derivative test
An example of looking for local minima in a multivariable function by finding where tangent planes are flat, along with some of the intuitions that will underly the second partial derivative test.
Lecture 83
Second partial derivative test
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Second partial derivative test
How to determine if the critical point of a two-variable function is a local minimum, a local maximum, or a saddle point.
Lecture 84
Second partial derivative test intuition
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Second partial derivative test intuition
The second partial derivative test is based on a formula which seems to come out of nowhere. Here, you can see a little more intuition for why it looks the way it does.
Lecture 85
Second partial derivative test example, part 1
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Second partial derivative test example, part 1
A worked example of finding a classifying critical points of a two-variable function.
Lecture 86
Second partial derivative test example, part 2
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Second partial derivative test example, part 2
Continuing the worked example from the previous video, now classifying each critical point.