Slope and equation of tangent line

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  1. Connecting: Derivative & Slope & Equation of Tangent Line
    Exercise: The graph of the quadratic function f(x)=12x2+2x−1f\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
  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)=2x3−3x2−12x+8f(x)=2x^3-3x^2-12x+8 where the tangent is horizontal.
    2. Find the vertex of each quadratic function:
  3. Locating Tangent Lines Parallel to a Linear Function
    Consider the Cubic function: f(x)=x3−3x2+3xf(x)=x^3-3x^2+3x
    i) Find the points on the curve where the tangent lines are parallel to the line 12x−y−9=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)=x2−x−2f(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:
      1. Sketch each parabola.
      2. Determine the lines that are tangent to both curves.
    Topic Notes
    The derivative of a function is interpreted as the slope of the tangent line to the curve of the function at a certain given point. In this section, we will explore the meaning of a derivative of a function, as well as learning how to find the slope-point form of the equation of a tangent line, as well as normal lines, to a curve at multiple given points.
    Point-Slope Form of a line with slope m through a point (x1,y1):m=y−y1x−x1(x_1,y_1): m=\frac{y-y_1}{x-x_1}

    Tangent Line & Normal Line
    The normal line to a curve at a particular point is the line through that point and perpendicular to the tangent line.