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homework-02

# homework-02 - (A 1 Ā 1.1 4g 2 Ā 1.1 4h 3 Prove for all n...

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Modern Algebra and Geometry I Math 4000/6000 (92-321) Homework Assignments 2A and 2B – Due Friday, January 13 Homework is due at the start of class. Work that is not turned in at the beginning of class is considered late, and will not generally be accepted. Write your solutions in complete mathematical sentences. Always provide a complete justiﬁcation for each solution, even if a problem does not explicitly ask you to do so. When using a result from the textbook as part of a solution, clearly identify the result you are using, e.g., “. . . so the statement is true for all n N by the Principle of Mathematical Induction,” or “. . . and 34 7 34 (mod 7) by Proposition 3.3 of Chapter 1.” This assignment consists of two parts that must be turned in separately. Problems in Part A will be marked by the course grader. Problems in Part B will be marked by the instructor.
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Unformatted text preview: (A) 1. Ā§ 1.1 # 4g 2. Ā§ 1.1 # 4h 3. Prove for all n ā N that ā n i =0 (4 i + 3) = (2 n + 3)( n + 1). 4. (a) State Rolleās Theorem from diļ¬erential Calculus. (b) Let a and b be real numbers. Prove for all n ā N that if f : R ā R is continuous on the interval [ a,b ], diļ¬erentiable on the interval ( a,b ), and has n roots in the interval [ a,b ], then the derivative f has at least n-1 roots in the interval [ a,b ]. (B) 1. Ā§ 1.1 # 11 (donāt forget to prove uniqueness) 2. Ā§ 1.1 # 17 3. Challenge problem: Ā§ 1.1 # 18 (If you follow the hint, are you inducting on k or on n ?) Additional practice problems: Ā§ 1.1 # 3, 4abcdehij, 5, 7 These problems will not be collected, but cover material that you will be responsible for knowing. 1...
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