Tuesday, April 19 (outline)
Coordinates and Functions.
- 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.
Section 10.0 - ON-Line Quiz
- Functions rule relating variables
- function notation x -> f(x)
Tutorial on Functions (numerical & algebraic)
Tutorial on Functions (Visual/Graphical)
Static vs. Dynamic visualizations
Winplot: a tool for visualizing functions
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
Four Problems Connecting the visual to the Numerical
- Motion and distance traveled by a falling object
- 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
- 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).