pra-140b-1

Pra-140b-1 - c ∈ a,b such that f p q 1 c = 0 8 Assume that ϕ is a monotonically increasing function and differentiable Assume further that for

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Math 140, Winter Practice problems April, 2010 Instructor: Professor Ni 1. Page 114: Exercise 4, Exercise 6. 2. Page 115: Exercise 11, 15. 3. Page 117: Exercise 22. 4. Page 138: Exercise 5. 5. Let f ( x ) = 1 - x 2 3 . Check that f ( - 1) = f (1) = 0. Explain why the mean value theorem fails to conclude that there exists c ( - 1 , 1) such that f 0 ( c ) = 0. 6. Assume that f and its derivatives f ( j ) are continuous on [ x 0 ,x n ] for all j n - 1 and f ( n ) exists on ( x 0 ,x n ). Assume that there exist x j with x 0 < x 1 < x 2 ··· < x n - 1 < x n such that f ( x 0 ) = f ( x 1 ) = ··· f ( x n ) . Prove that there exists c ( x 0 ,x n ) such that f ( n ) ( c ) = 0. 7. Let f is a function such that f ( j ) for j p + q exist on [ a,b ] and f ( p + q +1) exists on ( a,b ) and f ( a ) = f 0 ( a ) = ··· f ( p ) ( a ) = 0 and f ( b ) = f 0 ( b ) = ··· = f ( q ) ( b ) = 0 . Prove that there exists
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Unformatted text preview: c ∈ ( a,b ) such that f ( p + q +1) ( c ) = 0. 8. Assume that ϕ is a monotonically increasing function and differentiable. Assume further that for x ≥ x , | f ( x ) | ≤ ϕ ( x ). Prove that for x ≥ x | f ( x )-f ( x ) | ≤ ϕ ( x )-ϕ ( x ) . 9. Page 138: 4, 5, 6, 7, 8 10. Prove Theorem 6.12 on page 128. 11. Assume that α is monotonically increasing and α ∈ R , namely Riemann integrable, on [ a,b ]. Prove that for any bounded function f prove that ¯ Z b a f ( x ) dα = ¯ Z b a f ( x ) α ( x ) dx. 1...
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This note was uploaded on 09/30/2011 for the course MATH 140b taught by Professor Staff during the Winter '08 term at UCSD.

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