review_ch3 - Review Chapter 3 Section 3.1 Maximum and...

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Review Chapter 3 Section 3.1 Maximum and Minmum Values Definition: (i) f has an absolute maximum (or global maximum ) at “ c ” if f ( c ) f ( x ) for all x in D, where D is the domain of f . (ii) f has an absolute minimum (or global minimum ) at “ c ” if f ( c ) f ( x ) for all x in D, where D is the domain of f . (iii) f has an local maximum (or relative maximum ) at “ c ” if f ( c ) f ( x ) when x is near c. (iv) f has an local minimum (or relative minimum ) at “ c ” if f ( c ) f ( x ) when x is near c. (v) The maximum and minimum values of f are called the extreme values of f . Problem 1. State the absolute and local maximum and minimum values of the function from the graph. a b c d e f g a b c d e f g h (1) (2) Fermat’s Theorem If f has a local maximum or minimum at c and if f 0 ( c ) exists, then f 0 ( c ) = 0. 1
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Problem 2. Find the critical number(s) of the given function f . (1) f ( x ) = 4 x 3 - 3 x 2 - 6 x + 100 (2) f ( x ) = x - 2 x 2 - 1 (3) f ( x ) = cos x - sin 2 x, 0 < x < 2 π (4) f ( x ) = | x 2 + 4 x - 12 | Theorem: If f has a local maximum or minimum at c , then c is a critical number of f . The Closed Interval Method: To find the absolute maximum and minimum values of a continuous function f on a closed interval [ a, b ], Step 1. Find the values of f at the critical numbers of f in ( a, b ) Step 2. Find the values of f at the endpoints of the interval. Step 3. The largest of the values from step 1 and 2 is the absolute maximum value ; the smallest of these values is the absolute minimum value . Problem 3. Find the absolute maximum and absolute minimum values of f on the given interval. (1) f ( x ) = x 3 - 3 x 2 - 24 x - 2 , [ - 3 , 1] (2) f ( x ) = x 2 x 2 + 4 , [ - 1 , 1] Section 3.2 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 0 ( c ) = 0. 2
Problem 4. For what value of k will Rolle’s Theorem apply for the function f ( x ) = x 3 + kx - 5 on the interval [ - 1 , 3]? Problem 5. Given f ( x ) = x 2 + x + 1, find the number x = c where the conclusion of the Rolle’s Theorem is satisfied on the interval [ - 1 , 0]. The Mean Value Theorem Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval [ a, b ]. 2. f is differentiable on the open interval ( a, b ). Then there is a number c in ( a, b ) such that f 0 ( c ) = f ( b ) - f ( a ) b - a . Problem 6. Suppose that c = 1 satisfies the conclusion of the Mean Value Theorem for f ( x ) = 3 x 3 - kx + 1 on [ - 1 , 2]. Find k . Problem 7. Given f ( x ) = x x + 1 , find the number x = c where the conclusion of the Mean Value Theorem is satisfied on the interval [0 , 5]. Problem 8. Suppose that f ( - 1) = 3 and - 2 f 0 ( x ) 5 for all real numbers. What is the smallest possible value of f (7)? what is the largest possible value of f (7)? Section 3.3 How Derivatives Affect the Shape of a Graph Increasing/Decreasing Test (a) If f 0 ( x ) > 0 on an interval I , then f is increasing on I . (b) If f 0 ( x ) < 0 on an interval I , then f