# Konsultimet Mat avancuar Funksioni periodik Ushtrime te zgjidhura dhe udhezime.

1.Gjeni perioden e funksioneve te meposhtme: a)$y=sinx\cdot&space;cosx$   b)$y=2cos^{2}x-1$ , c)$y=sin^{2}x$   d)$y=\frac{cosx}{sinx}$

zgjidhje

Funksionet sinkx, coskx, kane perioden $T=\frac{2\pi&space;}{k}$, ndersa funksionet tgkx, cotgkx  kane perioden $T=\frac{\pi&space;}{k}$

a) $y=sinx\cdot&space;cosx=\frac{1}{2}\cdot&space;2sinx\cdot&space;cosx=\frac{1}{2}sin2x\Rightarrow&space;T=\frac{2\pi&space;}{2}=\pi$

b)$y=2cos^{2}x-1=cos^{2}x-sin^{2}x=cos2x\Rightarrow&space;T=\pi$

c)$cos2x=cos^{2}x-sin^{2}x=1-2sin^{2}x\Rightarrow&space;sin^{2}x=\frac{1}{2}\left&space;(&space;1-cos2x&space;\right&space;)$   Te vertetojme se perioda e ketij funksioni eshte pi.  Shenojme me T perioden e funksionit.$f(x+T)=f(x)\Rightarrow&space;\frac{1}{2}\left&space;[&space;1-cos2(x+T)&space;\right&space;]=\frac{1}{2}\left&space;(&space;1-cos2x&space;\right&space;)\Leftrightarrow&space;1-cos2(x+T)=1-cos2x\Leftrightarrow&space;cos2(x+T)=cos2x\Rightarrow&space;2(x+T)-2x=2\pi&space;\Rightarrow&space;T=\pi$

d)Njelloj funksioni eshte cotgx.

2.Vertetoni se nese funksioni f eshte periodik me periode T, atehere funksioni g:$y=A\cdot&space;f(kx+b)$  ku A, k, b jane konstante (k>0) eshte periodik me periode T/k.

zgjidhje

Funksioni f periodik kjo do te thote se $f(x+T)=f(x)\Rightarrow&space;f\left&space;[&space;(kx+b)+T&space;\right&space;]=f(kx+b)$ . E zeme se a eshte perioda e funksionit g d.m.th $g(x+a)=g(x)\Rightarrow&space;g(x+a)=A\cdot&space;f\left&space;[&space;k(x+a)+b&space;\right&space;]=A\cdot&space;f\left&space;[&space;(kx+b)+ka&space;\right&space;]=A\cdot&space;f(kx+b)=g(x)\Rightarrow&space;ka=T\Rightarrow&space;a=\frac{T}{k}$

Postimi tjeter per funksionet jo periodike