# 0209 - Math 31A 2010.02.09 MATH 31A DISCUSSION JED YANG...

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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 Rolle’s 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 4.3.53–55. 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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0209 - Math 31A 2010.02.09 MATH 31A DISCUSSION JED YANG...

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