Tuesday,  April 19 (outline)

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Parabolas

• Line of symmetry
• Coordinate graphs of quadratics relation
• Finding line of symmetry from data, from equation
• Secant lines and slopes
• Locatring the vertex. and slopes of secant lines.
  

Coordinates and Functions.

• Functions rule relating variables
• function notation x -> f(x)
  

Section 10.0 - ON-Line Quiz 
Tutorial on Functions (numerical & algebraic)  
Tutorial on Functions (Visual/Graphical)
Static vs. Dynamic visualizations
Winplot: a tool for visualizing functions

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Ch. 10.1 Some Historical Problems of Visualization:

• [Optional- Read:The parabola and squares.
• Ratio of Length of verticals :: ratio of squares on horizontals]
  
• Visualizing Algebra, Motion and Change
• time vs. position
• time vs velocity
  
• [Optional- Read:Analytic geometry- Descartes and Fermat]
• Section 10.1 - Quiz

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10.2  Four Problems Connecting the visual to the Numerical

• Motion and distance traveled by a falling object
• Visualize:
  
• constant velocity connected to area of rectangle
  
• constantly increasing velocity connected to area of a triangle.
  
• Motion and position (cannon balls)
• Visual connections:
• Horizontal motion- constant velocity.
• Vertical motion- constantly increasing velocity.
• Combined- parabola!
  
• Tangent line:
  
• Visualize: Position and velocity.
  
• Linear position, velocity is slope.
• Quadratic position, velocity is slope of tangent.
• Estimation of slope
  
• From graphing
• From slopes of Secant lines
  
• Area of a region
• Squares and Rectangles
• Triangles
• Circles (estimations and Archimedes)
• Estimations for  parabolic region. 
•  Section 10.2 - Quiz
• 10.3, 10.4 Newton/Cauchy:
• Newton: Tangent lines, velocity, and the "derivative."
• Finding slope of  tangent (newton/cauchy) 
• Finding "instantaneous velocity"
• Abstraction of common numerical measurement procedure:
  
• The derivative is a number which is
  
• determined at "the limit" of numbers corresponding to slopes of secants lines with short bases (of length dx) and/or
  
• average velocities with short time intervals (of duration dt).
  

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