DOC PREVIEW
NCSU MA 242 - 6e_challprobs

This preview shows page 1-2-3-4-5 out of 16 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 16 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

CHALLENGE PROBLEMS1Stewart: Calculus,Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.CHAPTER 31. (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 41. Find the absolute maximum value of the function2. (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 theangle 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).4. Let and be positive numbers. Show that not both of the numbers and can be greater than .5. Let be a triangle with and .(a) Express the length of the angle bisector in terms of .(b) Find the largest possible value of .CHAPTER 51. Show that .2. Suppose the curve passes through the origin and the point . Find the value ofthe integral .3. 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 wederive formulas for any . These formulas were first published in 1713 by the Swiss mathe-matician James Bernoulli in his book Ars Conjectandi.(a) The Bernoulli polynomials are defined by , , andfor . Find for and .(b) Use the Fundamental Theorem of Calculus to show that for .n 艌 2Bn共0兲 苷 Bn共1兲4n 苷 1, 2, 3,Bn共x兲n 苷 1, 2, 3, . . .x10 Bn共x兲 dx 苷 0Bn⬘共x兲 苷 Bn⫺1共x兲B0共x兲 苷 1Bnkk 苷 1, 2,nkx10 f ⬘共x兲 dx共1, 1兲y 苷 f 共x兲117艋y21 11 ⫹ x4 dx 艋724ⱍADⱍx 苷ⱍABⱍADⱍABⱍⴢⱍACⱍ苷 1⬔BAC 苷 120⬚ABC14b共1 ⫺ a兲a共1 ⫺ b兲bada兾dtdc兾dtdb兾dt␪cbd␪兾dt␪a␪tca, br␪a␪ar␪苷12⬔CⱍCDⱍ苷12(ⱍBCⱍ⫹ⱍACⱍ⫺ⱍABⱍ)Da 苷ⱍBCⱍAABCf 共x兲 苷11 ⫹ⱍxⱍ⫹11 ⫹ⱍx ⫺ 2ⱍf ⬘ff ⬘共x兲f 共x兲 苷s1 ⫺s2 ⫺s3 ⫺ x Click here for answers.AClick here for solutions.SClick here for answers.AClick here for solutions.SClick here for answers.AClick here for solutions.SBACDFIGURE FOR PROBLEM 2(c) If we introduce the Bernoulli numbers , then we can writeand, in general,where[The numbers are the binomial coefficients.] Use part (b) to show that, for ,and thereforeThis gives an efficient way of computing the Bernoulli numbers and therefore theBernoulli 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 younotice 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 forthe sum of the first cubes in Section 5.2.(j) Show that the formula in part (h) can be written symbolically aswhere the expression is to be expanded formally using the Binomial Theorem and each power is to be replaced by the Bernoulli number .(k) Use part (j) to find a formula for .equator that have exactlythe same temperature.CHAPTER 61. A solid is generated by rotating about the -axis the region under the curve , whereis a positive function and . The volume generated by the part of the curve fromto is for all . Find the function .fb ⬎ 0b2x 苷 bx 苷 0x 艌 0fy 苷 f 共x兲x15⫹ 25⫹ 35⫹⭈⭈⭈⫹n5bibi共n ⫹ 1 ⫹ b兲k⫹11k⫹ 2k⫹ 3k⫹⭈⭈⭈⫹nk苷1k ⫹ 1 关共n ⫹ 1 ⫹ b兲k⫹1⫺ bk⫹1兴nB4k 苷 31k⫹ 2k⫹ 3k⫹⭈⭈⭈⫹ nk苷 k! 关Bk⫹1共n ⫹ 1兲 ⫺ Bk⫹1共0兲兴 苷 k! yn⫹10 Bk共x兲 dxx 苷 0, 1, 2, ..., nBk⫹1共x ⫹ 1兲 ⫺ Bk⫹1共x兲 苷 xk兾k!0 艋 x 艋 1B1, B2, ..., B9B9B8B7B6B5b8b6n ⬎ 0b2n⫹1苷 0Bn共1 ⫺ x兲 苷 共⫺1兲nBn共x兲bn⫺1苷 ⫺1n 冋冉n0冊b0⫹冉n1冊b1⫹冉n2冊b2⫹⭈⭈⭈⫹冉nn ⫺ 2冊bn⫺2册bn苷兺nk苷0 冉nk冊bkn 艌 2(nk)冉nk冊苷n!k!共n ⫺ k兲!Bn共x兲 苷1n! 兺nk苷0 冉nk冊bkxn⫺k B3共x兲 苷x33!⫹b11! x22!⫹b22! x1!⫹b33! B2共x兲 苷x22!⫹b11! x1!⫹b22! B1共x兲 苷x1!⫹b11! B0共x兲 苷 b0bn苷 n! Bn共0兲2 ■ CHALLENGE PROBLEMSStewart: Calculus,Sixth Edition. ISBN: 0495011606. © 2008 Brooks/Cole. All rights reserved.Click here for answers.AClick here for solutions.SCHAPTER ;1. The Chebyshev polynomials are defined by , , , , ,....(a) What are the domain and range of these functions?(b) We know that and . Express explicitly as a quadratic polynomialand as a cubic polynomial.(c) Show that, for , .(d) Use part (c) to show that is a polynomial of degree .(e) Use parts (b) and (c) to express and explicitly as polynomials.(f) What are the zeros of ? At what numbers does have local maximum and minimumvalues?(g) Graph , , , and on a common screen.(h) Graph , , and on a common screen.(i) Based on your observations from parts (g) and (h), how are the zeros of related to thezeros of ? What about the -coordinates of the maximum and minimum values?(j) Based on your graphs in parts (g) and (h), what can you say about when isodd and when is even?(k) Use the substitution to evaluate the integral in part (j).(l) The family of functions are defined even when is not an integer(but then is not a polynomial). Describe how the graph of changes as increases.CHAPTER 1. A circle of radius has its center at the origin. A circle of radius rolls without slipping inthe counterclockwise direction around . A point is located on a fixed radius of the rollingcircle at a distance from its center, . [See parts (i) and (ii) of the figure.] Let bethe line from the center of to the center of the rolling circle and let be the angle that makes with the positive -axis.(a) Using as a parameter, show that parametric equations of the path traced out by are,. Note: If , the path is a circle ofradius ; if , the path is an epicycloid. The path traced out by for iscalled an epitrochoid.;(b) Graph the curve for various values of between and .(c) Show that an equilateral triangle can be inscribed in the epitrochoid and that its centroid ison the circle of radius centered at the origin.Note: This is the principle of the Wankel rotary engine. When the


View Full Document

NCSU MA 242 - 6e_challprobs

Download 6e_challprobs
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view 6e_challprobs and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view 6e_challprobs 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?