Slope and equation of tangent line

Lessons

• Introduction
  Connecting: Derivative & Slope & Equation of Tangent Line
  Exercise: The graph of the quadratic function $f\left( x \right) = \frac{1}{2}{x^2} + 2x - 1$ is shown below.
  
  
  a)

  Find and interpret $f'\left( x \right)$.
  
  
  b)

  Find the slope of the tangent line at:
  i) $x = - 1$
  ii) $x = 2$
  iii) $x = - 7$
  iv) $x = - 4$
  v) $x = - 2$
  
  
  c)

  Find an equation of the tangent line at:
  i) $x = 2$
  ii) $x = - 4$
  iii) $x = - 2$
  
  
  

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• 1.
  Determining Equations of the Tangent Line and Normal Line
  Consider the function: $f(x)=\frac{x}{32}(\sqrt{x}+{^3}\sqrt{x})$
  a)

  Determine an equation of the tangent line to the curve at $x=64$.
  
  
  b)

  Determine an equation of the normal line to the curve at $x=64$.
  
  

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• 2.
  Locating Horizontal Tangent Lines
  a)

  Find the points on the graph of $f(x)=2x^3-3x^2-12x+8$ where the tangent is horizontal.
  
  
  b)

  Find the vertex of each quadratic function:
  $f(x)=2x^2-12x+10$
  $g(x)=-3x^2-60x-50$
  
  

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• 3.
  Locating Tangent Lines Parallel to a Linear Function
  Consider the Cubic function: $f(x)=x^3-3x^2+3x$
  i) Find the points on the curve where the tangent lines are parallel to the line $12x-y-9=0$.
  ii) Determine the equations of these tangent lines.

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• 4.
  Determining Lines Passing Through a Point and Tangent to a Function
  Consider the quadratic function: $f(x)=x^2-x-2$
  a)

  Draw two lines through the point (3, -5) that are tangent to the parabola.
  
  
  b)

  Find the points where these tangent lines intersect the parabola.
  
  
  c)

  Determine the equations of both tangent lines.
  
  

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• 5.
  Locating Lines Simultaneously Tangent to 2 Curves
  Consider the quadratic functions:
  $f(x)=x^2$
  $g(x)=\frac{1}{4}x^2+3$
  a)

  Sketch each parabola.
  
  
  b)

  Determine the lines that are tangent to both curves.
  
  

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