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Unit: Applications of multivariable derivatives

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Tangent planes and local linearization

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What is a tangent plane
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Controlling a plane in space
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Computing a tangent plane
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Local linearization
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Tangent planes
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Local linearization
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Practice
Finding tangent planesGet 3 of 4 questions to level up!

Quadratic approximations

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What do quadratic approximations look like
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Quadratic approximation formula, part 1
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Quadratic approximation formula, part 2
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Quadratic approximation example
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The Hessian matrix
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Expressing a quadratic form with a matrix
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Vector form of multivariable quadratic approximation
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The Hessian
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Quadratic approximation
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Practice
The Hessian matrixGet 3 of 4 questions to level up!

Quiz 1

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Optimizing multivariable functions

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Multivariable maxima and minima
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Saddle points
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Warm up to the second partial derivative test
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Second partial derivative test
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Second partial derivative test intuition
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Second partial derivative test example, part 1
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Second partial derivative test example, part 2
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Practice
Find critical points of multivariable functionsGet 3 of 4 questions to level up!
Visual zero gradientGet 3 of 4 questions to level up!
Classifying critical pointsGet 3 of 4 questions to level up!

Optimizing multivariable functions (articles)

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Maxima, minima, and saddle points
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Second partial derivative test
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Reasoning behind second partial derivative test
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Examples: Second partial derivative test
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Gradient descent
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Quiz 2

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Lagrange multipliers and constrained optimization

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Constrained optimization introduction
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Lagrange multipliers, using tangency to solve constrained optimization
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Finishing the intro lagrange multiplier example
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Lagrange multiplier example, part 1
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Lagrange multiplier example, part 2
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The Lagrangian
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Meaning of the Lagrange multiplier
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Proof for the meaning of Lagrange multipliers
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Constrained optimization (articles)

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Lagrange multipliers, introduction
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Lagrange multipliers, examples
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Interpretation of Lagrange multipliers
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Unit test

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About this unit

The tools of partial derivatives, the gradient, etc. can be used to optimize and approximate multivariable functions. These are very useful in practice, and to a large extent this is why people study multivariable calculus.