︠73a659ea-c9e0-435c-bb21-a272a13d0a2di︠
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<h2>
Za pomoć pri rješavanju pogledajte primjere sa prošla dva labosa!
</h2>
︡16022e9a-df31-45cd-989c-73421fcabafd︡{"done":true,"html":"<h2>\nZa pomoć pri rješavanju pogledajte primjere sa prošla dva labosa!\n</h2>"}
︠64cb1fa8-2d94-4caf-af03-cf6f5bc94122i︠
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<h3>Zadatak 1</h3>
<p>Nacrtajte graf funkcije $\newcommand{\arcsin}{\mathop{\rm arcsin}\nolimits}f(x)=2-2\arccos(3x-1)$ na cijeloj prirodnoj domeni.</p>

︡ffa866fe-daff-4b3c-b3bc-d805567a6f55︡{"done":true,"html":"\n<h3>Zadatak 1</h3>\n<p>Nacrtajte graf funkcije $\\newcommand{\\arcsin}{\\mathop{\\rm arcsin}\\nolimits}f(x)=2-2\\arccos(3x-1)$ na cijeloj prirodnoj domeni.</p>"}
︠7725c1e6-571c-4209-86ae-29f60a89cad4︠
x=var('x')
f(x) = 2-2*arccos(3*x-1)
show(f)
plot(f)
︡08aa8c2e-3381-412d-bd23-3a38f8c72ca9︡{"html":"<div align='center'>$\\displaystyle x \\ {\\mapsto}\\ -2 \\, \\arccos\\left(3 \\, x - 1\\right) + 2$</div>"}︡{"stdout":"verbose 0 (3748: plot.py, generate_plot_points) WARNING: When plotting, failed to evaluate function at 134 points.\n"}︡{"stdout":"verbose 0 (3748: plot.py, generate_plot_points) Last error message: ''\n"}︡{"file":{"filename":"/home/user/.sage/temp/project-a04ef66b-8ef8-4dd6-9c89-bb7134d80c90/344/tmp_iEMiIk.svg","show":true,"text":null,"uuid":"e1ee40d5-9a30-4d4b-a906-5b19675a4dfe"},"once":false}︡{"done":true}︡
︠92cc958a-b05b-473b-9e54-3ff1432f3253i︠
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<h3>Zadatak 2</h3>
<p>Odredite Taylorov polinom 7. stupnja $T_5(x)$ pripadan funkciji $f(x)=\newcommand{\ch}{\mathop{\rm ch}\nolimits}\sqrt{\ch(x)}$ oko točke $x_0=0$. Nacrtajte grafove funkcija $f(x)$ i $T_5(x)$ na intervalu $[-5,5]$. Grafove crtajte u istom koordinatnom sustavu i različitim bojama. Koristite naredbu <a href="http://www.sagemath.org/doc/constructions/calculus.html#power-series"><span style="font-family: 'courier new', courier;">taylor</span></a>.</p>

︡84e5bb34-b643-4228-a7c0-c15aa85fbf65︡{"done":true,"html":"<h3>Zadatak 2</h3>\n<p>Odredite Taylorov polinom 7. stupnja $T_5(x)$ pripadan funkciji $f(x)=\\newcommand{\\ch}{\\mathop{\\rm ch}\\nolimits}\\sqrt{\\ch(x)}$ oko točke $x_0=0$. Nacrtajte grafove funkcija $f(x)$ i $T_5(x)$ na intervalu $[-5,5]$. Grafove crtajte u istom koordinatnom sustavu i različitim bojama. Koristite naredbu <a href=\"http://www.sagemath.org/doc/constructions/calculus.html#power-series\"><span style=\"font-family: 'courier new', courier;\">taylor</span></a>.</p>"}
︠afc92ee4-19d4-4722-b3c0-3debbf95e3e9︠
x=var('x')
f(x) = sqrt(cosh(x))

P=plot(f,-5,5, color='darkblue')

T=f.taylor(x,0,7)
view("$T_5(x) = $", T)

P+=plot(T,-5,5, color='lime')
show(P)
︡0073895c-64ae-4077-ad2b-1c555e115969︡{"html":"<div align='center'>$T_5(x) = $ $\\displaystyle \\frac{19}{5760} \\, x^{6} - \\frac{1}{96} \\, x^{4} + \\frac{1}{4} \\, x^{2} + 1$</div>"}︡{"file":{"filename":"/home/user/.sage/temp/project-a04ef66b-8ef8-4dd6-9c89-bb7134d80c90/417/tmp_5Uurik.svg","show":true,"text":null,"uuid":"b7d23623-ac73-414c-be8b-a182b4c92bea"},"once":false}︡{"done":true}︡
︠b9107d81-000e-4a9f-a95d-582a0a3edddei︠
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<h3>Zadatak 3</h3>
<p>Neka je $f(x)=\sin(x^3)$. Za interaktivno zadani cijeli broj $h\in[0,50]$, na jednom grafu istovremeno različitim bojama nacrtajte funkcije $g(x)=f'(x)$ i $g_h(x)={1\over2}{f(x+2^{-h})-f(x-2^{-h})\over 2^{-h}}$, a na drugom grafu funkciju $g_\delta(x)=g(x)-g_h(x)$. Oba grafa neka budu na intervalu $[0,2.5]$. &Scaron;to se zbiva kada je $h$ velik?</p>

︡41d18884-a455-4bba-af19-09bdb4a5ee58︡{"done":true,"html":"<h3>Zadatak 3</h3>\n<p>Neka je $f(x)=\\sin(x^3)$. Za interaktivno zadani cijeli broj $h\\in[0,50]$, na jednom grafu istovremeno različitim bojama nacrtajte funkcije $g(x)=f'(x)$ i $g_h(x)={1\\over2}{f(x+2^{-h})-f(x-2^{-h})\\over 2^{-h}}$, a na drugom grafu funkciju $g_\\delta(x)=g(x)-g_h(x)$. Oba grafa neka budu na intervalu $[0,2.5]$. &Scaron;to se zbiva kada je $h$ velik?</p>"}
︠e8c6fefd-335f-4a98-a6bf-fa3295f889f3s︠
x=var('x')
f(x) = sin(x^3)
g(x) = diff(f, x)
@interact
def _(h=(0,50,1)):
    print h
    # Naš kod:
    gh(x) = (f(x+2^(-h))-f(x-2^(-h)))/(2*2^(-h))
    gd(x) = g(x)-gh(x)

    view("$f(x) = $", f, "(nije na grafu)")
    view("$g(x) = $", g, "(plavo)")
    view("$g_h(x) = $", gh, "(žuto)")

    P1=plot(g,0,2.5, color='darkblue')
    P1+=plot(gh,0,2.5, color='darkorange')
    show(P1)

    view("$gd(x) =$", gd, "(tirkizno)")

    P2=plot(gd,0,2.5, color='cyan')
    show(P2)

︡e0b0dc25-5324-426a-9d50-a26d5cbea6a8︡{"interact":{"controls":[{"animate":true,"control_type":"slider","default":0,"display_value":true,"label":"h","vals":["0","1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50"],"var":"h","width":null}],"flicker":false,"id":"3b9a11ce-4603-4956-ad51-c13a582fb0f1","layout":[[["h",12,null]],[["",12,null]]],"style":"None"}}︡{"done":true}︡
︠cc9f0b82-664c-4db1-9e65-a096c40777bfi︠
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<h3>Zadatak 4</h3>
<p>Odredite jednu primitivnu funkciju funkcije $f(x)=4x^2-5x+4\sin(2x)$ te izračunajte određeni integral funkcije $f$ na intervalu $[-5,5]$. Koristite naredbu <a href="http://www.sagemath.org/doc/reference/calculus/sage/symbolic/integration/integral.html"><span style="font-family: 'courier new', courier;">integrate</span></a>.</p>
︡403ea672-c30b-441d-9a21-1bb94d024885︡{"done":true,"html":"<h3>Zadatak 4</h3>\n<p>Odredite jednu primitivnu funkciju funkcije $f(x)=4x^2-5x+4\\sin(2x)$ te izračunajte određeni integral funkcije $f$ na intervalu $[-5,5]$. Koristite naredbu <a href=\"http://www.sagemath.org/doc/reference/calculus/sage/symbolic/integration/integral.html\"><span style=\"font-family: 'courier new', courier;\">integrate</span></a>.</p>"}
︠12680b1a-bad9-4fba-8665-9aa182b0b1b1s︠
x=var('x')
f(x) = 4*x^2-5*x+4*sin(2*x)
fprim=f.integrate(x)
view("Primitivna funkcija: ", fprim)
fint=integrate(f,x,-5,5)
show(fint)
︡180012af-86fd-40be-a2b2-6156e0470fea︡{"html":"<div align='center'>Primitivna funkcija:  $\\displaystyle x \\ {\\mapsto}\\ \\frac{4}{3} \\, x^{3} - \\frac{5}{2} \\, x^{2} - 2 \\, \\cos\\left(2 \\, x\\right)$</div>"}︡{"html":"<div align='center'>$\\displaystyle \\frac{1000}{3}$</div>"}︡{"done":true}
︠da9ed237-2997-4e06-a72b-ccad34f917c0︠