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WebChallengeProblems_5E

WebChallengeProblems_5E - CHALLENGE PROBLEMS 1 Challenge...

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|||| Chapter 3 1. (a) Find the domain of the function . (b) Find . ; (c) Check your work in parts (a) and (b) by graphing and on the same screen. |||| Chapter 4 1. Find the absolute maximum value of the function 2. (a) Let be a triangle with right angle and hypotenuse . (See the figure.) If the inscribed circle touches the hypotenuse at , show that (b) If , express the radius of the inscribed circle in terms of and . (c) If is fixed and varies, find the maximum value of . 3. A triangle with sides , and varies with time , but its area never changes. Let be the angle opposite the side of length and suppose always remains acute. (a) Express in terms of , , , , and . (b) Express in terms of the quantities in part (a). |||| Chapter 5 1. In Sections 5.1 and 5.2 we used the formulas for the sums of the th powers of the first integers when and 3. (These formulas are proved in Appendix E.) In this problem we derive for- mulas for any . These formulas were first published in 1713 by the Swiss mathematician James Bernoulli in his book Ars Conjectandi . (a) The Bernoulli polynomials are defined by , , and for . Find for and . (b) Use the Fundamental Theorem of Calculus to show that for . (c) If we introduce the Bernoulli numbers , then we can write and, in general, where [The numbers are the binomial coefficients.] Use part (b) to show that, for , b n n k 0 n k b k n 2 ( n k ) n k n ! k ! n k ! B n x 1 n ! n k 0 n k b k x n k B 3 x x 3 3! b 1 1! x 2 2! b 2 2! x 1! b 3 3! B 2 x x 2 2! b 1 1! x 1! b 2 2! B 1 x x 1! b 1 1! B 0 x b 0 b n n ! B n 0 n 2 B n 0 B n 1 4 n 1, 2, 3, B n x n 1, 2, 3, . . . x 1 0 B n x dx 0 B n x B n 1 x B 0 x 1 B n k k 1, 2, n k da dt dc dt db dt c b d dt a t c a , b r a a r 1 2 C CD 1 2 ( BC AC AB ) D a BC A ABC f x 1 1 x 1 1 x 2 f f f x f x s 1 s 2 s 3 x CHALLENGE PROBLEMS 1 Challenge Problems B A C D FIGURE FOR PROBLEM 2 Click here for answers. A Click here for solutions. S Click here for answers. A Click here for solutions. S Click here for answers. A Click here for solutions. S
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and therefore This gives an efficient way of computing the Bernoulli numbers and therefore the Bernoulli polynomials. (d) Show that and deduce that for . (e) Use parts (c) and (d) to calculate and . Then calculate the polynomials , , , , and . ; (f) Graph the Bernoulli polynomials for . What pattern do you notice in the graphs? (g) Use mathematical induction to prove that . (h) By putting in part (g), prove that (i) Use part (h) with and the formula for in part (a) to confirm the formula for the sum of the first cubes in Section 5.2. (j) Show that the formula in part (h) can be written symbolically as where the expression is to be expanded formally using the Binomial and each power is to be replaced by the Bernoulli number . (k) Use part (j) to find a formula for . |||| Chapter 6 1. A solid is generated by rotating about the -axis the region under the curve , where is a positive function and . The volume generated by the part of the curve from to is for all . Find the function .
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