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Unformatted text preview: page 1 © Li Chen , Spring 2008 Chapter 4 Chapter 4. Applications of Differentiation 4.1. Maximum and Minimum Values Absolute Maximum and Absolute Minimum • f has an absolute minimum at c if f ( c ) ≤ f ( x ) for all x in D • f has an absolute maximum at c if f ( c ) ≥ f( x ) for all x in D where D is the domain of f ( x ). Local Maximum and Local Minimum We say that f ( c ) is a local maximum if f ( c ) ≥ f ( x ) for all x near c . We say that f ( c ) is a local minimum if f ( c ) ≤ f ( x ) for all x near c . This means that f ( c ) ≥ f ( x ) (or f ( c ) ≤ f ( x )) for all x in some open interval containing c. page 2 © Li Chen , Spring 2008 Chapter 4 Ex 1. Find the absolute and local maximum and minimum values for the function whose graph is shown below. 2 2 22 page 3 © Li Chen , Spring 2008 Chapter 4 The Extreme Value Theorem If a function is continuous on a closed interval [ a , b ], then the function must have both an absolute maximum and an absolute minimum. a b a b page 4 © Li Chen , Spring 2008 Chapter 4 Definition A point x = c is a critical point if 1. c is in the domain of the function f ( x ), and 2. f ′ ( c ) = 0 or f ′ ( c ) does not exist. Ex 2. Find the absolute maximum and minimum values of the function ] 3 , 1 [ 2 9 6 ) ( 2 3 + + = x x x x f page 5 © Li Chen , Spring 2008 Chapter 4 Ex 3. (a) Sketch the graph of a function that has two local maximum, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minimum, two local maximum, and seven critical numbers. page 6 © Li Chen , Spring 2008 Chapter 4 Ex 4. Find the critical numbers of the function. 2 2 1 ) ( ) a ( x xe x f = ) 8 ( ) ( ) b ( 3 t t t g = page 7 © Li Chen , Spring 2008 Chapter 4 4.2 The Mean Value Theorem Rolle’s Theorem Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [ a, b ]. 2. f is differentiable on the open interval ( a, b ). 3. f ( a ) = f ( b ) Then there is a number c in ( a, b ) such that f ´( c ) = 0. 2 0 2 4 22 page 8 © Li Chen , Spring 2008 Chapter 4 Ex 1. Verify that the given function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. ] 3 , [ 2 6 ) ( 2 3 + = x x x x f page 9 © Li Chen , Spring 2008 Chapter 4 The Mean Value Theorem If f is a differentiable function on the interval [a, b], then there exists a number c between a and b such that ( ) ( ) '( ) f b f a f c b a = Ex 2 . Use the graph of f below to estimate the value of c that satisfies the conclusion of the Mean Value Theorem for the interval [3, 5]....
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 Spring '08
 Don'tremeber
 Calculus, Critical Point, Derivative, Li Chen

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