Kafli 7

$x=2+\frac{3}{y+1},$ eða $y=\mathrm{-1}+\frac{3}{x-2}.$ Þessi jafna lýsir hluta réttstæðs breiðboga með miðju í $\left(2,\mathrm{-1}\right).$

Einn möguleiki er $x\left(t\right)=t,y\left(t\right)=t^2+2t.$ Annar möguleiki er $x\left(t\right)=2t-3,y\left(t\right)={\left(2t-3\right)}^2+2\left(2t-3\right)=4t^2-8t+3.$

$x^{\prime }\left(t\right)=2t-4$ og $y^{\prime }\left(t\right)=6t^2-6,$ og því $\frac{dy}{dx}=\frac{6t^2-6}{2t-4}=\frac{3t^2-3}{t-2}.$
Þessi segð er óskilgreind þegar $t=2$ og jöfn núlli þegar $t=\mathrm{\pm 1}.$

Jafna snertilsins er $y=24x+100.$

$\frac{d^{2}y}{d{x}^2}=\frac{3t^2-12t+3}{2{\left(t-2\right)}^3}.$ Markpunktar $\left(5,4\right),\left(\mathrm{-3},\mathrm{-4}\right),\text{og}\left(\mathrm{-4},4\right).$

$A=3\pi$ (Athugið að heildisformúlan gefur í raun neikvætt svar. Ástæðan er sú að $x\left(t\right)$ er minnkandi fall á bilinu $\left[0,2\pi \right];$ það er að ferillinn er rakinn frá hægri til vinstri.)

$s=2\left({10}^{3\text{/}2}-2^{3\text{/}2}\right)\approx 57.589$

$A=\frac{\pi \left(494\sqrt{13}+128\right)}{1215}$

$\left(8\sqrt{2},\frac{5\pi }{4}\right)$ og $\left(\mathrm{-2},2\sqrt{3}\right)$

$y=x^2,$ sem er jafna fleygboga sem opnast upp.

$A=3\pi \text{/}2$

$A=\frac{4\pi }{3}+2\sqrt{3}$

$s=3\pi$

$x=2{\left(y+3\right)}^2-2$

$\frac{{\left(x+1\right)}^2}{16}+\frac{{\left(y-2\right)}^2}{9}=1$

$\frac{{\left(y+2\right)}^2}{9}-\frac{{\left(x-1\right)}^2}{4}=1.$ Þetta er lóðréttur breiðbogi. Aðfellur $y=\mathrm{-2}\pm \frac{3}{2}\left(x-1\right).$

$e=\frac{c}{a}=\frac{\sqrt{74}}{7}\approx 1.229$

Hér er $e=0.8$ og $p=5.$ Þetta keilusnið er sporbaugur.

Keilusniðið er breiðbogi og snúningshorn ásanna er $\theta =22.5\text{°}.$

$y=\frac{x^2}{4}+1$

Aðfellurnar eru $y=x$ og $y=\text{−}x$

$y=\frac{\sqrt{x+1}}{2}$; skilgreiningarmengi: $x\in \left[\mathrm{–1},\infty \right).$

$\frac{x^2}{16}+\frac{y^2}{9}=1;$ skilgreiningarmengi $x\in \left[\mathrm{-4},4\right].$

$y=3x+2;$ skilgreiningarmengi: allar rauntölur.

${(x-1)}^2+{(y-3)}^2=1;$ skilgreiningarmengi: $x\in \left[0,2\right].$

$y=\sqrt{x^2-1};$ skilgreiningarmengi: $x\in \left(\mathrm{-\infty },-1\right].$

$y^2=\frac{1-x}{2};$ skilgreiningarmengi: $x\in [-1,1].$

$y=\text{ln}x;$ skilgreiningarmengi: $x\in [1,\infty ).$

$y=\text{ln}x;$ skilgreiningarmengi: $x\in (0,\infty ).$

$x^2+y^2=4;$ skilgreiningarmengi: $x\in [\mathrm{-2},2].$

$\frac{\mathrm{-3}}{5}$

$\text{Hallatala}=0;$ $y=8.$

Hallatalan er óskilgreind; $x=2.$

$\tan t = -2(\frac{4}{\sqrt{5}},\frac{\mathrm{-8}}{\sqrt{5}}), (\frac{4}{\sqrt{5}},\frac{\mathrm{-8}}{\sqrt{5}})$, $\left(\frac{-4}{\sqrt{5}},\frac{8}{\sqrt{5}}\right)$.

$y=\text{−}\left(\frac{4}{e}\right)x+5$

$y=\mathrm{–2}x+3$

$\frac{\pi }{4},\frac{5\pi }{4},\frac{3\pi }{4},\frac{7\pi }{4}$

$\frac{dy}{dx}=\text{−}\text{tan}(t)$

$\frac{dy}{dx}=\frac{3}{4}$ og $\frac{d^{2}y}{d{x}^2}=0,$ og því er ferillinn hvorki kúptur upp né niður í $t=3.$ Grafið er því línulegt og hefur fasta hallatölu en enga kúptni.

$\frac{dy}{dx}=4,\frac{d^{2}y}{d{x}^2}=\mathrm{-6}\sqrt{3};$ ferillinn er kúptur niður í $\theta =\frac{\pi }{6}.$

Engir láréttir snertlar. Lóðréttir snertlar í $(1,0),(\mathrm{-1},0).$

$\text{−}{\text{sec}}^3\left(\pi t\right)$

Láréttir $\left(0,\mathrm{-9}\right);$ lóðréttir $\left(\text{±}2,\mathrm{-6}\right).$

Kúpt upp þegar $t>0.$

$\frac{e^{\frac{\mathrm{\pi }}{2}}–1}{2}$

$\frac{3\pi }{2}$

$6\pi a^2$

$2\pi ab$

$\frac{1}{3}\left(2\sqrt{2}-1\right)$

$7.075$

$6a$

$6\sqrt{2}$

$\frac{2\pi \left(247\sqrt{13}+64\right)}{1215}$

$\frac{8\pi }{3}\left(17\sqrt{17}-1\right)$

$B\begin{array}{cc}\left(3,\frac{\text{−}\pi }{3}\right)\hfill & B\left(\mathrm{-3},\frac{2\pi }{3}\right)\hfill \end{array}$

$D\left(5,\frac{7\pi }{6}\right)D\left(\mathrm{-5},\frac{\pi }{6}\right)$

$(5, 5.356) (–5, 2.214)$

$(10, 2.214)(–10, 5.356)$

$\left(2\sqrt{3},\frac{11\mathrm{\pi }}{6}\right)\left(\mathrm{-2}\sqrt{3},\frac{5\mathrm{\pi }}{6}\right)$

$\left(\begin{array}{cc}\text{−}\sqrt{3},\hfill & \mathrm{-1}\hfill \end{array}\right)$

$\left(\begin{array}{cc}-\frac{\sqrt{3}}{2},\hfill & \frac{\mathrm{-1}}{2}\hfill \end{array}\right)$

$\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)$

Samhverfa um x-ásinn, y-ásinn og upphafspunktinn.

Aðeins samhverft um x-ásinn.

Aðeins samhverfa um x-ásinn.

Lína $y=x$

$y=1$

Breiðbogi; skauthnitaform $r^2\text{cos}(2\theta )=16$ eða $r^2=16\text{sec}(2\theta ).$

$r=\frac{2}{3\text{cos}\theta -\text{sin}\theta }$

$x^2+y^2=4y$

$x\text{tan}\sqrt{x^2+y^2}=y$

samhverfa um y-ásinn

samhverfa um y-ásinn

samhverfa um x- og y-ásana og um skautið

samhverfa um x-ásinn

samhverfa um x- og y-ásana og um skautið

$\frac{9}{2}{\displaystyle {\int }_0^{\pi }{\text{sin}}^2\theta d\theta }$

$32{\displaystyle {\int }_0^{\pi \text{/}2}{\text{sin}}^2(2\theta )d\theta }$

$\frac{1}{2}{\displaystyle {\int }_{\pi }^{2\pi }{\left(1-\text{sin}\theta \right)}^{2}d\theta }$

${\displaystyle {\int }_{{\text{sin}}^{\mathrm{-1}}\left(2\text{/}3\right)}^{\pi \text{/}2}{\left(2-3\text{sin}\theta \right)}^2}d\theta$

${\displaystyle {\int }_{\pi \text{/}3}^{\pi }{\left(1-2\text{cos}\theta \right)}^{2}d\theta -{\displaystyle {\int }_0^{\pi \text{/}3}{\left(1-2\text{cos}\theta \right)}^2}}d\theta$

$4{\displaystyle {\int }_0^{\pi \text{/}3}d\theta +16{\displaystyle {\int }_{\pi \text{/}3}^{\pi \text{/}2}\left({\text{cos}}^2\theta \right)}}d\theta$

$9\pi$

$\frac{9\pi }{4}$

$\frac{9\pi }{8}$

$\frac{18\pi -27\sqrt{3}}{2}$

$\frac{4}{3}\left(4\pi -3\sqrt{3}\right)$

$\frac{3}{2}\left(4\pi -3\sqrt{3}\right)$

$2\pi -4$

${\displaystyle {\int }_0^{2\pi }\sqrt{{\left(1+\text{sin}\theta \right)}^2+{\text{cos}}^2\theta }}d\theta$

$\sqrt{2}{\displaystyle {\int }_0^1{e}^{\theta }d\theta }$

$\frac{\sqrt{10}}{3}\left(e^6-1\right)$

$A=\pi {\left(\frac{\sqrt{2}}{2}\right)}^2=\frac{\pi }{2}\text{og}\frac{1}{2}{\displaystyle {\int }_0^{\pi }\left(1+2\text{sin}\theta \text{cos}\theta \right)}d\theta =\frac{\pi }{2}$

$C=2\pi \left(\frac{3}{2}\right)=3\pi \text{og}{\displaystyle {\int }_0^{\pi }3}d\theta =3\pi$

$C=2\pi \left(5\right)=10\pi \text{og}{\displaystyle {\int }_0^{\pi }10}d\theta =10\pi$

$\frac{dy}{dx}=\frac{f^{\prime }(\theta )\text{sin}\theta +f\left(\theta \right)\text{cos}\theta }{f^{\prime }\left(\theta \right)\text{cos}\theta -f(\theta )\text{sin}\theta }$

Hallatalan er $\frac{1}{\sqrt{3}}.$

Í $\left(4,0\right),$ er hallatalan óskilgreind. Í $\left(\mathrm{-4},\frac{\pi }{2}\right),$ er hallatalan 0.

Hallatalan er óskilgreind í $\frac{4+\mathrm{\pi }}{4-\mathrm{\pi }}$

Hallatalan er $\frac{\mathrm{-2}}{\pi }.$

Láréttur snertill í $\left(\text{±}\sqrt{2},\frac{\pi }{6}\right),$ $\left(\text{±}\sqrt{2},-\frac{\pi }{6}\right).$

Láréttir snertlar í $\frac{\pi }{2},\frac{7\pi }{6},\frac{11\pi }{6}.$ Lóðréttir snertlar í $\frac{\pi }{6},\frac{5\pi }{6}$ og einnig í skautinu $(0,0).$

$y^2=16x$

$x^2=2y$

$x^2=\mathrm{-4}\left(y-3\right)$

${\left(x+3\right)}^2=8\left(y-3\right)$

$\frac{x^2}{16}+\frac{y^2}{12}=1$

$\frac{x^2}{13}+\frac{y^2}{4}=1$

$\frac{{\left(y-1\right)}^2}{16}+\frac{{\left(x+3\right)}^2}{12}=1$

$\frac{x^2}{16}+\frac{y^2}{12}=1$

$\frac{x^2}{25}-\frac{y^2}{11}=1$

$\frac{x^2}{7}-\frac{y^2}{9}=1$

$\frac{{\left(y+2\right)}^2}{4}-\frac{{\left(x+2\right)}^2}{32}=1$

$\frac{x^2}{4}-\frac{y^2}{32}=1$

$e=1,$ fleygbogi

$e=\frac{1}{2},$ sporbaugur

$e=3,$ breiðbogi

$r=\frac{4}{5+\text{cos}\theta }$

$r=\frac{4}{1+2\text{sin}\theta }$

$r=\frac{2.616}{1+0.995\text{cos}\theta }$

$r=\frac{5.192}{1+0.0484\text{cos}\theta }$

Ósatt. Skoðið $y=t+1,$ $x=\text{−}t+1.$

$y=1-x^3$

$\frac{x^2}{16}+{(y-1)}^2=1$

$r^2=\frac{4}{{\text{sin}}^2\theta -{\text{cos}}^2\theta }$

$y=\frac{3\sqrt{2}}{2}+\frac{1}{5}\left(x+\frac{3\sqrt{2}}{2}\right)$

$\frac{e^2}{2}$

$9\sqrt{10}$

${\left(y+5\right)}^2=\mathrm{-8}x+32$

$\frac{{\left(y+1\right)}^2}{16}-\frac{{\left(x+2\right)}^2}{9}=1$

$e=\frac{2}{3},$ sporbaugur

$\frac{y^2}{{19.03}^2}+\frac{x^2}{{19.63}^2}=1,$ $e=0.2447$