mat25-hw5solns

mat25-hw5solns - Math 25 Advanced Calculus UC Davis Spring...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Solutions to homework assignment #5 Reading material. Read sections 2.4–2.7 in the textbook. Problems on previous material 1. Answer the following problems in the textbook: A.7.1, A.7.3, A.8.2, A.8.5, 1.9.2, 1.9.6, 1.10.3, 1.10.5 Solution to A.7.1. Let n = 3. For this n , the equation 4 x 2 + x - n = 0 becomes 4 x 2 + x - 3 = 0. This equation has a rational root x = - 1. Solution to A.7.3. The converse: “Every continuous function is diﬀerentiable” (this is false — e.g., the function f ( x ) = | x | is continu- ous but not diﬀerentiable). The contrapositive: “Every function that is not continuous is also not diﬀerentiable”. This is true. Solution to A.8.2. I claim that 1 + 3 + 5 + ... + (2 n - 1) = n 2 . The proof is by induction on n . For n = 1, the left-hand side is 1, and n 2 = 1, so the statement is true. Assume that it is true for n - 1. This means that 1 + 3 + 5 + ... + ( 2( n - 1) - 1 ) = 1 + 3 + 5 + ... + (2 n - 3) = ( n - 1) 2 . It follows, using the induction hypothesis, that 1 + 3 + 5+ ... + (2 n - 1) = (1 + 3 + 5 + ... + (2 n - 3)) + (2 n - 1) = ( n - 1) 2 + (2 n - 1) = n 2 - 2 n + 1 + (2 n - 1) = n 2 . Solution to 1.10.5.

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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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mat25-hw5solns - Math 25 Advanced Calculus UC Davis Spring...

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