Thursday, April 21 (outline)
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.
- 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).
- Formula for the deriavative as a number and as a function!
- Graphical relation: the derivative as a function and its graph
- Reversing the derivative and the Fundamental Theorem.
- 10.5, 10.6
Determining position and areas.