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Applications of multivariable derivatives

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What is a tangent planeControlling a plane in spaceComputing a tangent planeLocal linearizationTangent planesLocal linearization
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What do quadratic approximations look likeQuadratic approximation formula, part 1Quadratic approximation formula, part 2Quadratic approximation exampleThe Hessian matrixExpressing a quadratic form with a matrixVector form of multivariable quadratic approximationThe HessianQuadratic approximation
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Multivariable maxima and minimaSaddle pointsWarm up to the second partial derivative testSecond partial derivative testSecond partial derivative test intuitionSecond partial derivative test example, part 1Second partial derivative test example, part 2
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Maxima, minima, and saddle pointsSecond partial derivative testReasoning behind second partial derivative testExamples: Second partial derivative test
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Constrained optimization introductionLagrange multipliers, using tangency to solve constrained optimizationFinishing the intro lagrange multiplier exampleLagrange multiplier example, part 1Lagrange multiplier example, part 2The LagrangianMeaning of the Lagrange multiplierProof for the meaning of Lagrange multipliers
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Lagrange multipliers, introductionLagrange multipliers, examplesInterpretation of Lagrange multipliers

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.