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.