A.7.3: Section 7.3 Answers
- Page ID
- 43778
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \(y = 2 − 3x − 2x^{2} + \frac{7}{2}x^{3} − \frac{55}{12}x^{4} + \frac{59}{8}x^{5} − \frac{83}{6}x^{6} + \frac{9547}{ 336}x^{7} + \ldots\)
2. \(y = −1 + 2x − 4x^{3} + 4x^{4} + 4x^{5} − 12x^{6} + 4x^{7} + \ldots \)
3. \(y = 1 + x^{2} − \frac{2}{3}x^{3} + \frac{11}{6}x^{4} − \frac{9}{5} x^{5} + \frac{329}{90}x^{6} − \frac{1301}{315} x^{7} + \dots\)
4. \(y = x − x^{2} − \frac{7}{2} x^{3} + \frac{15}{2} x^{4} + \frac{45}{8} x^{5} − \frac{261}{8}x^{6} + \frac{207}{16} x^{7} + \ldots\)
5. \(y = 4 + 3x − \frac{15}{4} x^{2} + \frac{1}{4} x^{3} + \frac{11}{16} x^{4} − \frac{5}{16} x^{5} + \frac{1}{20} x^{6} + \frac{1}{120} x^{7} + \ldots\)
6. \(y = 7 + 3x − \frac{16}{3} x^{2} + \frac{13}{3} x^{3} − \frac{23}{9} x^{4} + \frac{10}{9} x^{5} − \frac{7}{27} x^{6} − \frac{1}{9} x{7} + \ldots\)
7. \(y = 2 + 5x − \frac{7}{4} x^{2} − \frac{3}{16} x^{3} + \frac{37}{192} x^{4} − \frac{7}{192} x^{5} − \frac{1}{1920} x^{6} + \frac{19}{11520} x^{7} + \ldots\)
8. \(y = 1 − (x − 1) + \frac{4}{3} (x − 1)^{3} − \frac{4}{3} (x − 1)^{4} − \frac{4}{5} (x − 1)^{5} + \frac{136}{45} (x − 1)^{6} − \frac{104}{63} (x − 1)^{7} + \ldots \)
9. \(y = 1 − (x + 1) + 4(x + 1)^{2} − \frac{13}{3} (x + 1)^{3} + \frac{77}{6} (x + 1)^{4} − \frac{278}{15} (x + 1)^{5} + \frac{1942}{45} (x + 1)^{6} − \frac{23332}{315} (x + 1)^{7} + \ldots\)
10. \(y = 2 − (x − 1) − \frac{1}{2} (x − 1)^{2} + \frac{5}{3} (x − 1)^{3} − \frac{19}{12} (x − 1)^{4} + \frac{7}{30} (x − 1)^{5} + \frac{59}{45} (x − 1)^{6} − \frac{1091}{630} (x − 1)^{7} + \ldots\)
11. \(y = −2 + 3(x + 1) − \frac{1}{2}(x + 1)^{2} − \frac{2}{3}(x + 1)^{3} + \frac{5}{8}(x + 1)^{4} − \frac{11}{30} (x + 1)^{5} + \frac{29}{144} (x + 1)^{6} − \frac{101}{840 }(x + 1)^{7} + \ldots\)
12. \(y = 1 − 2(x − 1) − 3(x − 1)^{2} + 8(x − 1)^{3} − 4(x − 1)^{4} − \frac{42}{5}(x − 1)^{5} + 19(x − 1)^{6} − \frac{604}{35} (x − 1)^{7} + \ldots\)
19. \(y = 2 − 7x − 4x^{2} − \frac{17}{6}x^{3} − \frac{3}{4}x^{4} − \frac{9}{40}x^{5} + \ldots\)
20. \(y = 1 − 2(x − 1) + \frac{1}{2} (x − 1)^{2} − \frac{1}{6}(x − 1)^{3} + \frac{5}{36}(x − 1)^{4} − \frac{73}{1080}(x − 1)^{5} + \ldots\)
21. \(y = 2 − (x + 2) −\frac{7}{2}(x + 2)^{2} +\frac{4}{3}(x + 2)^{3} −\frac{1}{24}(x + 2)^{4} +\frac{1}{60}(x + 2)^{5} + \ldots\)
22. \(y = 2 − 2(x + 3) − (x + 3)^{2} + (x + 3)^{3} −\frac{11}{12}(x + 3)^{4} +\frac{67}{60}(x + 3)^{5} +\ldots\)
23. \(y = −1 + 2x + \frac{1}{3}x^{3} −\frac{5}{12}x^{4} + \frac{2}{5}x^{5} + \ldots\)
24. \(y = 2 − 3(x + 1) + \frac{7}{2} (x + 1)^{2} − 5(x + 1)^{3} + \frac{197}{24}(x + 1)^{4} − \frac{287}{20}(x + 1)^{5} + \ldots\)
25. \(y = −2 + 3(x + 2) − \frac{9}{2}(x + 2)^{2} +\frac{11}{6}(x + 2)^{3} +\frac{5}{24}(x + 2)^{4} +\frac{7}{20}(x + 2)^{5} + \ldots\)
26. \(y = 2 − 4(x − 2) − \frac{1}{2}(x − 2)^{2} + \frac{2}{9}(x − 2)^{3} + \frac{49}{432}(x − 2)^{4} + \frac{23}{1080} (x − 2)^{5} + \ldots\)
27. \(y = 1 + 2(x + 4) − \frac{1}{6}(x + 4)^{2} −\frac{10}{27}(x + 4)^{3} +\frac{19}{648}(x + 4)^{4} +\frac{13}{324}(x + 4)^{5} + \ldots\)
28. \(y = −1 + 2(x + 1) −\frac{1}{4}(x + 1)^{2} +\frac{1}{2}(x + 1)^{3} −\frac{65}{96}(x + 1)^{4} + \frac{67}{80}(x + 1)^{5} + \ldots\)
31.
- \(y=\frac{c_{1}}{1+x}+\frac{c_{2}}{1+2x}\)
- \(y=\frac{c_{1}}{1-2x}+\frac{c_{2}}{1-3x}\)
- \(y=\frac{c_{1}}{1-2x}+\frac{c_{2}x}{(1-2x)^{2}}\)
- \(y=\frac{c_{1}}{2+x}+\frac{c_{2}x}{(2+x)^{2}}\)
- \(y=\frac{c_{1}}{2+x}+\frac{c_{2}}{2+3x}\)
32. \(y = 1 − 2x −\frac{3}{2} x^{2} + \frac{5}{3} x^{3} + \frac{17}{24} x^{4} − \frac{11}{20} x^{5} + \ldots\)
33. \(y = 1 − 2x − \frac{5}{2} x^{2} + \frac{2}{3} x^{3} − \frac{3}{8} x^{4} + \frac{1}{3} x^{5} + \ldots\)
34. \(y = 6 − 2x + 9x^{2} + \frac{2}{3} x^{3} − \frac{23}{4} x^{4} − \frac{3}{10} x^{5} + \ldots\)
35. \(y = 2 − 5x + 2x^{2} − \frac{10}{3} x^{3} + \frac{3}{2} x^{4} − \frac{25}{12} x^{5} + \ldots\)
36. \(y = 3 + 6x − 3x^{2} + x^{3} − 2x^{4} − \frac{17}{20} x^{5} + \ldots\)
37. \(y = 3 − 2x − 3x^{2} + \frac{3}{2} x^{3} + \frac{3}{2} x^{4} − \frac{49}{80} x^{5} + \ldots\)
38. \(y = −2 + 3x + \frac{4}{3} x^{2} − x^{3} − \frac{19}{54} x^{4} + \frac{13}{60} x^{5} + \ldots\)
39. \(y_{1}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m}}{m!}=e^{-x^{2}},\:y_{2}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m+1}}{m!}=xe^{-x^{2}}\)
40. \(y = −2 + 3x + x^{2} − \frac{1}{6} x^{3} −\frac{3}{4}x^{4} + \frac{31}{120}x^{5} + \ldots\)
41. \(y = 2 + 3x − \frac{7}{2} x^{2} − \frac{5}{6} x^{3} + \frac{41}{24} x^{4} + \frac{41}{120} x^{5} + \ldots\)
42. \(y = −3 + 5x − 5x^{2} + \frac{23}{6}x^{3} − \frac{23}{12}x^{4} + \frac{11}{30} x^{5} + \ldots\)
43. \(y = −2 + 3(x − 1) + \frac{3}{2}(x − 1)^{2} −\frac{17}{12}(x − 1)^{3} −\frac{1}{12}(x − 1)^{4} +\frac{1}{8}(x − 1)^{5} + \ldots\)
44. \(y = 2 − 3(x + 2) + \frac{1}{2}(x + 2)^{2} − \frac{1}{3}(x + 2)^{3} + \frac{31}{24}(x + 2)^{4} −\frac{53}{120}(x + 2)^{5} + \ldots\)
45. \(y = 1 − 2x + \frac{3}{2} x^{2} − \frac{11}{6} x^{3} + \frac{15}{8} x^{4} − \frac{71}{60} x^{5} + \ldots\)
46. \(y = 2 − (x + 2) − \frac{7}{2}(x + 2)^{2} − \frac{43}{6}(x + 2)^{3} − \frac{203}{24} (x + 2)^{4} − \frac{167}{30} (x + 2)^{5} + \ldots\)
47. \(y = 2 − x − x^{2} + \frac{7}{6} x^{3} − x^{4} + \frac{89}{120} x^{5} + \ldots\)
48. \(y = 1 + \frac{3}{2} (x − 1)^{2} + \frac{1}{6} (x − 1)^{3} − \frac{1}{8} (x − 1)^{5} + \ldots\)
49. \(y = 1 − 2(x − 3) + \frac{1}{2}(x − 3)^{2} − \frac{1}{6}(x − 3)^{3} + \frac{1}{4}(x − 3)^{4} − \frac{1}{6}(x − 3)^{5} + \ldots\)