Ejercicios de repaso sección 5.1

Demuestre que las funciones son ortogonales en el intervalo indicado

  1. f1(x)=x      f2(x)=x2      [−2,2]\hspace{0.5cm} f_1(x)=x\;\;\;f_2(x)=x^2 \;\;\;\left[-2,2\right]

  2. f1(x)=x3      f2(x)=x2+1      [−1,1]\hspace{0.5cm} f_1(x)=x^3\;\;\;f_2(x)=x^2+1\;\;\;\left[-1,1\right]

  3. f1(x)=ex      f2(x)=xe−x−e−x      [0,2]\hspace{0.5cm} f_1(x)=e^x\;\;\;f_2(x)=xe^{-x}-e^{-x}\;\;\;\left[0,2\right]

  4. f1(x)=cosx      f2(x)=sen2x      [0,π]\hspace{0.5cm} f_1(x)=cos{x}\;\;\;f_2(x)=sen^2x\;\;\;\left[0,\pi\right]

  5. f1(x)=x      f2(x)=cos2x      [−π2,π2]\hspace{0.5cm} f_1(x)=x\;\;\;f_2(x)=cos{2}x\;\;\;\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]

  6. f1(x)=ex      f2(x)=senx      [π4,5π4]\hspace{0.5cm} f_1(x)=e^x\;\;\;f_2(x)=senx\;\;\;\left[\dfrac{\pi}{4},\dfrac{5\pi}{4}\right]

    7. En los ejercicios 7 a 12 demuestre que el conjunto dado de funciones es ortogonal en el intervalo indicado. Encuentre la norma de cada función en el conjunto

  7. {senx,sen3x,sen5x,...}      [0,π2]\hspace{0.5cm} \left\{senx,sen3x,sen5x,...\right\}\;\;\;\left[0,\dfrac{\pi}{2}\right]

  8. {cosx,cos3x,cos5x,...}      [0,π2]\hspace{0.5cm} \left\{cos{x},cos{3}x,cos{5}x,...\right\}\;\;\;\left[0,\dfrac{\pi}{2}\right]

  9. {sennx}\hspace{0.5cm} \left\{sennx\right\} con n=1,2,3,…      [0,π]n=1,2,3,\ldots\;\;\;\left[0,\pi\right]

  10. {sennπpx}\hspace{0.5cm} \left\{sen\frac{n\pi}{p}x\right\} con n=1,2,3,…      [0,p]1n=1,2,3,\ldots\;\;\;\left[0,p\right]1

  11. {1,cosnπpx}\hspace{0.5cm} \left\{1,cos{\dfrac{n\pi}{p}}x\right\} con n=1,2,3,…      [0,p]n=1,2,3,\ldots\;\;\;\left[0,p\right]

  12. {1,cosnπpx,senmπpx}      [−p,p]\hspace{0.5cm} \left\{1,cos{\dfrac{n\pi}{p}}x,sen\dfrac{m\pi}{p}x\right\}\;\;\;\left[-p,p\right]