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Section Solutions

Conics

  1. \(y = 4 (x-1)^2 + 3\text{,}\) directrix \(y=2\frac{15}{16}\text{,}\) focus \(= (1, 3\frac{1}{16})\text{,}\) vertex \(= (1,3)\text{,}\) and area \(= 1\frac{2}{3}\)

  2. center \(= (2,-1)\text{,}\) vertices \(= (2, -1 \pm \sqrt{6})\text{,}\) foci \(= (2, -1 \pm \sqrt{2})\text{,}\) minor extremities \(= (0,-1)\) and \((4,-1)\text{,}\) equation of tangent at \((0,-1)\) is \(x=0\)

  3. \(\dsp \frac{2}{3}\int_{-3}^3 \sqrt{1 + \frac{4}{9}(9-x^2)} \; dx = 5\invsin(\frac{2\sqrt{5}}{5}) \approx 7.54\)

  4. center \(= (2,-2)\text{,}\) vertices \(=(2,0)\) and \((2,-4)\text{,}\) asymptotes \(y = \pm \frac{2}{3}(x-2)-2\text{,}\) foci \(=(2, -2 \pm \sqrt{13})\)

  5. no solutions

Parametrics

    1. \(\dsp \frac{dy}{dx} = \frac{b \cos(t) }{-a \sin(t)}\text{,}\) \(\dsp \frac{d^2y}{dx^2} = \frac{b}{a^2 \sin^3(t)}\)

    2. \(\dsp \frac{dy}{dx} = \frac{\sqrt{1-t^2}}{1+t^2}\text{,}\) \(\dsp \frac{d^2y}{dx^2} = \frac{t(t^2-3)}{(1+t^2)^2}\)

    3. \(\dsp \frac{dy}{dx} = \frac{\ln(t) + 1}{t^{e-1}e^{t+1} + t^e e^t}\)

  2. \(\dsp y = -\frac{1}{2} x + 1\)

  3. stopped at \(t=0\text{,}\) moving left at all positive times, right at all negative times, \(y=-x+10\) for \(x \leq 6\)

  4. arc lengths

    1. \(\sqrt{2}(e^4-e^1)\)

    2. \(-6\)

  5. \(\dsp x = 1 \pm \sqrt{14/3}\)

  6. \(\dsp y=\frac{5}{3} x + 5\)

Polars

    1. \(\dsp r = \frac{7}{4\cos(\theta) - 3\sin(\theta)}\)

    2. \(\dsp r = \frac{1}{\pm \sqrt{\cos(\theta)\sin(\theta)}}\)

    3. \(\dsp r = \frac{3\cos(\theta)\sin(\theta)}{\cos^3(\theta)+\sin^3(\theta)}\)

    1. \(\dsp y = 2\sqrt{2x^2+3x+1}\)

    2. \(\dsp x^2 + y^2 = 3y\)

    3. \(\dsp x^4 + 2x^2y^2 + y^4 = 6x^2 - 6y^2\)

    4. \(x=5\)

    1. \(0 \pm \pi/6, \pi/2 \pm \pi/6, \pi \pm \pi/6, 3\pi/2 \pm \pi/6\)

    2. \(\pm \pi/3\)

    1. \(24\pi\)

    2. \(33\pi/2\)

    3. \(2\pi\)

    1. \(6\pi\)

    2. \(2(3\pi-8)/3\)

  6. \(\dsp \frac{\pi - 3\sqrt{3}}{2}\)

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