CHALLENGE PROBLEMS CHAPTER 3 A Click here for answers Click here for solutions S s1 s2 s 3 x b Find f x c Check your work in parts a and b by graphing f and f on the same screen 1 a Find the domain of the function f x CHAPTER 4 A Click here for answers Click here for solutions S 1 Find the absolute maximum value of the function f x 1 1 1 x 1 x 2 2 a Let ABC be a triangle with right angle A and hypotenuse a BC See the C figure If the inscribed circle touches the hypotenuse at D show that CD BC AC AB 1 2 D b If 12 C express the radius r of the inscribed circle in terms of a and c If a is fixed and varies find the maximum value of r A B 3 A triangle with sides a b and c varies with time t but its area never changes Let be the angle opposite the side of length a and suppose always remains acute a Express d dt in terms of b c db dt and dc dt b Express da dt in terms of the quantities in part a FIGURE FOR PROBLEM 2 4 Let a and b be positive numbers Show that not both of the numbers a 1 b and b 1 a can be greater than 14 5 Let ABC be a triangle with BAC 120 and AB AC 1 a Express the length of the angle bisector AD in terms of x AB b Find the largest possible value of AD Stewart Calculus Sixth Edition ISBN 0495011606 2008 Brooks Cole All rights reserved CHAPTER 5 A Click here for answers 1 Show that S Click here for solutions 1 2 1 7 y 4 dx 1 1 x 17 24 2 Suppose the curve y f x passes through the origin and the point 1 1 Find the value of the integral x01 f x dx 3 In Sections 5 1 and 5 2 we used the formulas for the sums of the k th powers of the first n integers when k 1 2 and 3 These formulas are proved in Appendix E In this problem we derive formulas for any k These formulas were first published in 1713 by the Swiss mathematician James Bernoulli in his book Ars Conjectandi a The Bernoulli polynomials Bn are defined by B0 x 1 Bn x Bn 1 x and x01 Bn x dx 0 for n 1 2 3 Find Bn x for n 1 2 3 and 4 b Use the Fundamental Theorem of Calculus to show that Bn 0 Bn 1 for n 2 1 2 CHALLENGE PROBLEMS c If we introduce the Bernoulli numbers bn n Bn 0 then we can write B0 x b0 B2 x x2 b1 x b2 2 1 1 2 B1 x x b1 1 1 B3 x x3 b1 x 2 b2 x b3 3 1 2 2 1 3 and in general Bn x 1 n n k 0 n bk x n k k where n k n k n k The numbers nk are the binomial coefficients Use part b to show that for n 2 n bn k 0 n bk k and therefore bn 1 d e f g h 1 n n b0 0 n b1 1 n b2 2 n bn 2 n 2 This gives an efficient way of computing the Bernoulli numbers and therefore the Bernoulli polynomials Show that Bn 1 x 1 nBn x and deduce that b2n 1 0 for n 0 Use parts c and d to calculate b6 and b8 Then calculate the polynomials B5 B6 B7 B8 and B9 Graph the Bernoulli polynomials B1 B2 B9 for 0 x 1 What pattern do you notice in the graphs Use mathematical induction to prove that Bk 1 x 1 Bk 1 x x k k By putting x 0 1 2 n in part g prove that 1k 2 k 3 k n k k Bk 1 n 1 Bk 1 0 k y n 1 0 Bk x dx i Use part h with k 3 and the formula for B4 in part a to confirm the formula for the sum of the first n cubes in Section 5 2 j Show that the formula in part h can be written symbolically as 1 n 1 b k 1 b k 1 k 1 where the expression n 1 b k 1 is to be expanded formally using the Binomial Theorem and each power b i is to be replaced by the Bernoulli number bi k Use part j to find a formula for 15 2 5 3 5 n 5 equator that have exactly the same temperature CHAPTER 6 A Click here for answers S Click here for solutions 1 A solid is generated by rotating about the x axis the region under the curve y f x where f is a positive function and x 0 The volume generated by the part of the curve from x 0 to x b is b 2 for all b 0 Find the function f Stewart Calculus Sixth Edition ISBN 0495011606 2008 Brooks Cole All rights reserved 1k 2 k 3 k n k CHALLENGE PROBLEMS 3 CHAPTER 8 A Click here for answers S Click here for solutions 1 The Chebyshev polynomials Tn are defined by Tn x cos n arccos x n 0 1 2 3 a What are the domain and range of these functions b We know that T0 x 1 and T1 x x Express T2 explicitly as a quadratic polynomial and T3 as a cubic polynomial c Show that for n 1 Tn 1 x 2x Tn x Tn 1 x d Use part c to show that Tn is a polynomial of degree n e Use parts b and c to express T4 T5 T6 and T7 explicitly as polynomials f What are the zeros of Tn At what numbers does Tn have local maximum and minimum values g Graph T2 T3 T4 and T5 on a common screen h Graph T5 T6 and T7 on a common screen i Based on your observations from parts g and h how are the zeros of Tn related to the zeros of Tn 1 What about the x coordinates of the maximum and minimum values 1 j Based on your graphs in parts g and h what can you say about x 1 Tn x dx when n is odd and when n is even k Use the substitution u arccos x to evaluate the integral in part j l The family of functions f x cos c arccos x are defined even when c is not an integer but then f is not a polynomial Describe how the graph of f changes as c increases CHAPTER 11 A Click here for answers S Click here for solutions 1 A circle C of radius 2r has its center at the origin A circle of radius r rolls without slipping in …