5.11.E: Problemas en Funciones Exponenciales y Trigonométricas
- Page ID
- 113990
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Verificar fórmula\((2)\).
\(\text { Prove Note } 1, \text { as suggested (using Chapter } 2, §§ 11-12)\).
Demostrar fórmulas\((1)\) de Capítulo\(2, §§11-12\) a partir de nuestras nuevas definiciones.
Completar los detalles faltantes en las pruebas de Teoremas\(2-4\).
Demostrar que
(i)\(\sin 0=\sin (n \pi)=0\);
(ii)\(\cos 0=\cos (2 n \pi)=1\);
(iii)\(\sin \frac{\pi}{2}=1\);
(iv)\(\sin \left(-\frac{\pi}{2}\right)=-1\);
(v)\(\cos \left( \pm \frac{\pi}{2}\right)=0\);
(vi)\(|\sin x| \leq 1\) y\(|\cos x| \leq 1\) para\(x \in E^{1}\).
Demostrar que
(i)\(\sin (-x)=-\sin x\) y
(ii)\(\cos (-x)=\cos x\) para\(x \in E^{1}\).
[Pista: Para (i), vamos\(h(x)=\sin x+\sin (-x) .\) Mostrar que\(h^{\prime}=0 ;\) por lo tanto\(h\) es constante, digamos,\(\left.h=q \text { on } E^{1} . \text { Substitute } x=0 \text { to find } q . \text { For (ii), use }(13)-(15) .\right]\)
Demostrar lo siguiente para\(x, y \in E^{1} :\)
(i) de\(\sin (x+y)=\sin x \cos y+\cos x \sin y ;\) ahí\(\sin \left(x+\frac{\pi}{2}\right)=\cos x\).
ii)\(\cos (x+y)=\cos x \cos y-\sin x \sin y ;\) por lo tanto\(\cos \left(x+\frac{\pi}{2}\right)=-\sin x\).
[Pista para\((\mathrm{i}) :\) Fijar\(x, y\) y dejar\(p=x+y .\) Definir\(h : E^{1} \rightarrow E^{1}\) por
\ [
h (t) =\ sin t\ cos (p-t) +\ cos t\ sin (p-t),\ quad t\ en E^ {1}.
\]
\(\text { Proceed as in Problem } 6 . \text { Then let } t=x .]\)
Con\(\overline{J_{n}}\) como en el texto, mostrar que el seno aumenta en\(\overline{J_{n}}\) si\(n\) es par y disminuye si\(n\) es impar. ¿Qué tal el coseno? Encuentra los puntos finales de\(\overline{J_{n}}\).