Slope and equation of tangent line

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Intros
Lessons
  1. Connecting: Derivative & Slope & Equation of Tangent Line
    Exercise: The graph of the quadratic function f(x)=12x2+2x1f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1 is shown below.
    Slope and equation of tangent line
  2. Find and interpret f(x)f'\left( x \right).
  3. Find the slope of the tangent line at:
    i) x=1x = - 1
    ii) x=2x = 2
    iii) x=7x = - 7
    iv) x=4x = - 4
    v) x=2x = - 2
  4. Find an equation of the tangent line at:
    i) x=2x = 2
    ii) x=4x = - 4
    iii) x=2x = - 2
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Examples
Lessons
  1. Determining Equations of the Tangent Line and Normal Line
    Consider the function: f(x)=x32(x+3x)f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})
    1. Determine an equation of the tangent line to the curve at x=64x=64.
    2. Determine an equation of the normal line to the curve at x=64x=64.
  2. Locating Horizontal Tangent Lines
    1. Find the points on the graph of f(x)=2x33x212x+8f(x)=2x^3-3x^2-12x+8 where the tangent is horizontal.
    2. Find the vertex of each quadratic function:
      f(x)=2x212x+10f(x)=2x^2-12x+10
      g(x)=3x260x50g(x)=-3x^2-60x-50
  3. Locating Tangent Lines Parallel to a Linear Function
    Consider the Cubic function: f(x)=x33x2+3xf(x)=x^3-3x^2+3x
    i) Find the points on the curve where the tangent lines are parallel to the line 12xy9=012x-y-9=0.
    ii) Determine the equations of these tangent lines.
    1. Determining Lines Passing Through a Point and Tangent to a Function
      Consider the quadratic function: f(x)=x2x2f(x)=x^2-x-2
      1. Draw two lines through the point (3, -5) that are tangent to the parabola.
      2. Find the points where these tangent lines intersect the parabola.
      3. Determine the equations of both tangent lines.
    2. Locating Lines Simultaneously Tangent to 2 Curves
      Consider the quadratic functions:
      f(x)=x2f(x)=x^2
      g(x)=14x2+3g(x)=\frac{1}{4}x^2+3
      1. Sketch each parabola.
      2. Determine the lines that are tangent to both curves.