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# Analyzing functions

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Mean value theoremMean value theorem example: polynomialMean value theorem example: square root functionJustification with the mean value theorem: tableJustification with the mean value theorem: equationEstablishing differentiability for MVTMean value theorem applicationMean value theorem review

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Using the mean value theoremGet 3 of 4 questions to level up!

Justification with the mean value theoremGet 3 of 4 questions to level up!

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Extreme value theoremCritical points introductionFinding critical points

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Find critical pointsGet 3 of 4 questions to level up!

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Finding decreasing interval given the functionFinding increasing interval given the derivativeIncreasing & decreasing intervals review

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Increasing & decreasing intervalsGet 3 of 4 questions to level up!

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Introduction to minimum and maximum pointsFinding relative extrema (first derivative test)Worked example: finding relative extremaAnalyzing mistakes when finding extrema (example 1)Analyzing mistakes when finding extrema (example 2)Finding relative extrema (first derivative test)Relative minima & maxima review

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Relative minima & maximaGet 3 of 4 questions to level up!

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Finding absolute extrema on a closed intervalAbsolute minima & maxima (entire domain)Absolute minima & maxima review

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Absolute minima & maxima (closed intervals)Get 3 of 4 questions to level up!

Absolute minima & maxima (entire domain)Get 3 of 4 questions to level up!

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Concavity introductionAnalyzing concavity (graphical)Inflection points introductionInflection points (graphical)

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Concavity introGet 3 of 4 questions to level up!

Inflection points introGet 3 of 4 questions to level up!

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Analyzing concavity (algebraic)Inflection points (algebraic)Mistakes when finding inflection points: second derivative undefinedMistakes when finding inflection points: not checking candidatesAnalyzing the second derivative to find inflection pointsConcavity reviewInflection points review

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Analyze concavityGet 3 of 4 questions to level up!

Find inflection pointsGet 3 of 4 questions to level up!

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Second derivative test

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Second derivative testGet 3 of 4 questions to level up!

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Curve sketching with calculus: polynomialCurve sketching with calculus: logarithmAnalyzing a function with its derivative

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Calculus based justification for function increasingJustification using first derivativeJustification using first derivativeInflection points from graphs of function & derivativesJustification using second derivative: inflection pointJustification using second derivative: maximum pointJustification using second derivativeConnecting f, f', and f'' graphicallyConnecting f, f', and f'' graphically (another example)

Practice

Justification using first derivativeGet 3 of 4 questions to level up!

Justification using second derivativeGet 3 of 4 questions to level up!

Connecting f, f', and f'' graphicallyGet 3 of 4 questions to level up!

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Optimization: sum of squaresOptimization: box volume (Part 1)Optimization: box volume (Part 2)Optimization: profitOptimization: cost of materialsOptimization: area of triangle & square (Part 1)Optimization: area of triangle & square (Part 2)Optimization problem: extreme normaline to y=x²Motion problems: finding the maximum acceleration

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OptimizationGet 3 of 4 questions to level up!

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Horizontal tangent to implicit curve

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Tangents to graphs of implicit relationsGet 3 of 4 questions to level up!

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Analyze functions (calculator-active)Get 3 of 4 questions to level up!