Thursday,  April 21 (outline)

Review:
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).
• 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.