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Unformatted text preview: Math 31A 2010.02.09 MATH 31A DISCUSSION JED YANG More Applications of the Derivative 1. MVT and Monotonicity 1.1. Mean Value Theorem. Assume that f is continuous on [ a, b ] and differen- tiable on ( a, b ). Then there exists a number c ( a, b ) such that f ( c ) = f ( b ) f ( a ) b a . In particular, if f ( a ) = f ( b ), we get Rolles Theorem. 1.2. Exercise 4.3.42. Show that f ( x ) = x 3 2 x 2 + 2 x is an increasing function. Solution. Notice f ( x ) = 3 x 2 4 x + 2. What is its minimum? Find its critical points: f ( x ) = 6 x 4, so x = 2 3 is the critical point. So f ( x ) has its minimum at x = 2 3 , which is f ( 2 3 ) = 2 3 . So f ( x ) > 0, thus f ( x ) is increasing. square 1.3. Exercise 4.3.5355. Prove that if f (0) = g (0) and f ( x ) g ( x ) for x 0, then f ( x ) g ( x ) for all x 0. Prove the following: (a) sin x x for x 0. (b) cos x 1 1 2 x 2 , (c) sin x x 1 6 x 3 , (d) cos x 1 1 2 x 2 + 1 24 x 4 ....
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