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1Chapter 3 Notes PacketCalculus AB – Cogswell Name _____________________________Sections and Topics: Section 3.1 – Extrema on an IntervalUnderstand the definition of extrema of a function on an intervalUnderstand the definition of relative extrema of a function on an open intervalFind extrema on a closed interval. Section 3.2 – Rolle’s Theorem and the Mean Value TheoremUnderstand and use Rolle’s TheoremUnderstand and use the Mean Value TheoremSection 3.3 – Increasing and Decreasing Functions and the First Derivative TestDetermine intervals on which a function is increasing or decreasingApply the First Derivative Test to find relative extrema of a functionSection 3.4 – Concavity and the Second Derivative TestDetermine intervals on which a function is concave upward or concave downwardFind any points of inflection of the graph of a functionApply the Second Derivative Test to find relative extrema of a functionSection 3.5 – A Summary of Curve SketchingAnalyze and sketch the graph of a functionSection 3.6 – Optimization ProblemsSolve applied minimum and maximum problemsSection 3.7 – DifferentialsUnderstand the concepts of a tangent line approximationCompare the value of the differential, dy, with the actual change in y, ∆ yEstimate a propagated error using a differentialFind the differential of a function using differentiation formulas
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3Section 3.1 – Extrema on an IntervalIn this lesson you will learn how to use a derivative to locate the minimum and maximum values of a function on a closed interval.Definition of ExtremaLet fbe defined on an interval Icontaining c.f(c)is the ________________________ of fon Iif f(c)≤f(x)for all xin I.f(c)is the ________________________ of fon Iif f(c)≥f(x)for all xin I.The minimum and maximum of a function on an interval are the extreme values, or extrema, of the function on the interval. The minimum and maximum of the function on an interval are also called the absolute minimumand absolute maximumon the interval. “Global” is sometimes used for “absolute.”The Extreme Value TheoremIf fis continuous on a closed interval [a, b], then fhas both a minimum and a maximum on the interval.Definition of Relative (local) Extrema1. If there is an open interval containing con which f(c)is a maximum, then f(c)is called a _______________________________________of f2. If there is an open interval containing con which f(c)is a minimum, then f(c)is called a ________________________________________of f..
horizontal tangentf '(c) = 0cc is a critical number of fcf '(c) does not existc is a critical number of fRelative MaximumRelative Minimumaba and b are critical numbers4Definition of a Critical NumberLet fbe defined at c. If f'(c)=0or if fis not differentiable at c, then c is a _____________________ of f. (It is important that you notice that fhas to be defined at c.)Relative Extrema Occur Only at Critical NumbersIf fhas a relative minimum or relative maximum at x = c, then cis a critical number of f.

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