Exploring the derivative
Author: Lenore Horner

We explore the concept of slope for functions other than a straight line.

1. Drag point A (brown diamond) along the blue function.  What do you observe?

2. Move points B and C so that they bracket A in different ways.  How do the two slopes compare?  What is going on here?

a\left(x-x_0\right)^3+b\left(x-x_0\right)^2+c\left(x-x_0\right)+d

a = -0.1 (slider from -5 to 5, step 0.1)
b = 0.2 (slider from -5 to 5, step 0.1)
c = 0 (slider from -5 to 5, step 0.1)
x_0 = 3.1 (slider from -5 to 5, step 0.1)
d = 1.8 (slider from -5 to 5, step 0.1)
f(x) = a * (x - x_0)^(3.0) + b * (x - x_0)^(2.0) + c * (x - x_0) + d
A = Point(f)
e = Tangent(A, f)
slope = Slope(e)
S: (x(A), Slope[e])
B = Point(f)
C = Point(f)
rise: y(C) - y(B)
run: x(C) - x(B)
text1: "rise = " + rise
text2: "run = " + run
text3: "slope BC = " + (rise / run)
deriv(x) = a * 3.0 * (x - x_0)^(2.0) + b * 2.0 * (x - x_0) + c
g = Segment(B, C)