MathStudyGuide - Math-1A(S3 Fall 2002 Calculus Midterm#2...

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Math-1A(S3) Fall 2002, Calculus Midterm #2 Review (Chapter 4), 11-03-02 Key Notes Concepts : absolute (global) maximum/minimum; local maximum/minimum; extreme values; critical numbers (points); inflection points; increasing/decreasing; concave upward/downward; asymptotes (horizontal, vertical, slant, etc.); Rolle’s Theorem and Mean Value Theorem; L’Hospital’s Rule; Newton’s Method; Antiderivative; Cost and Revenue in Economics 1 Extreme Values A function f has a global (absolute) maximum (minimum) at c if f ( c ) f ( x ) ( f ( c ) f ( x )) for all x in the domain of f . A function f has a local maximum (minimum) at c if f ( x ) f ( x ) ( f ( c ) f ( x )) for all x in some open interval containing c . Extreme Value Theorem. If f is continuous on a closed interval [ a, b ], then f attains a global (absolute) maximum value f ( c ) and a global (absolute) minimum value f ( d ) at some numbers c and d in [ a, b ]. Fermat’s Theorem. If f has a local maximum or minimum at c , and if f ( c ) exists, then f ( c ) = 0. A critical number of a function f is a number c in the domain of f such that either f ( c ) = 0 or f ( c ) does not exist. The Closed Interval Method To find the global (absolute) maximum and minimum: 1. Find critical numbers ( f ( c ) = 0 or f ( c ) does not exist), and the corresponding critical values f ( c ); 2. Find the values of f at the endpoints of the interval; 3. The largest of the values from Step 1 and 2 is the global (absolute) maximum and the smallest is the global minimum.
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