Chapter 3.4 Practice

Questions

  1. For each circle \(C\) and point \(p\) on \(C\), determine the equation of the line that is tangent to \(C\) at \(p\).
    1. \(C\) given by the equation \(x^2+y^2=9\), \(p=\left(\frac{9}{5},\frac{12}{5}\right)\)
    2. \(C\) given by the equation \((x+2)^2+y^2=10\), \(p=\left(-1,3\right)\)
    3. \(C\) given by the equation \(x^2+(y-4)^2=5\), \(p=\left(1,6\right)\)
    4. \(C\) given by the equation \((x-2)^2+(y-\tfrac{1}{2})^2=82\), \(p=\left(-7,\tfrac{3}{2}\right)\)
  2. For each quadratic function \(f\), and point \(p\), identify a quadratic equation in \(m\) that determines the slope \(m\) of \(L\) so that it is tangent to \(f\) at \(p\).
    1. \(f(x)=5x^2-3x-4\), \(p=(-2,22)\)
    2. \(f(x)=-2x^2+x-4\), \(p=(-1,1)\)
    3. \(f(x)=5x^2+3x\), \(p=(1,8)\)
  3. For each quadratic function \(f\), and point \(p\), determine the line \(L\) that is tangent to \(f\) at \(p\).
    1. \(f(x)=5x^2-3x-4\), \(p=(-2,22)\)
    2. \(f(x)=-2x^2+x-4\), \(p=(-1,1)\)
    3. \(f(x)=5x^2+3x\), \(p=(1,8)\)
  4. For each polynomial function \(f\), and point \(p\), determine the line \(L\) that is tangent to \(f\) at \(p\).
    1. \(f(x)=x^3+5x^2+x+2\), \(p=(1,9)\)
    2. \(f(x)=3x^4-3x+1\), \(p=(1,1)\)
    3. \(f(x)=-x^4+2x^2-1\), \(p=(2,-9)\)

Answers

    1. \(y=-\frac{3}{4}x-\frac{15}{4}\) \(p=\left(\frac{9}{5},\frac{12}{5}\right)\)
    2. \(y=-\frac{1}{3}x+\frac{8}{3}\)
    3. \(y=-\frac{1}{2}x+\frac{13}{2}\)
    4. \(y=9x+\frac{129}{2}\)
    1. \(m^2-46m+529=0\)
    2. \(m^2+10m+25=0\)
    3. \(m^2+26m+169=0\)
    1. \(y=-23x-24\)
    2. \(y=5x-2\)
    3. \(y=13x-5\)
    1. \(y=14x-5\)
    2. \(y=9x-8\)
    3. \(y=-40x+55\)

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