Rate of Change of Volume of a Cube
Author: John Maharry

Visually explore the rate of change of V = x³

Let 'h' be the increase in the length of the sides of a cube.

Change in Volume = (x+h)³-x³
                                   = 3x²h + 3xh²+h³
                                   =( 3 'faces'  + 3 'edges' + 1 'corner' )

So the Rate of Change of the Volume is 

3x²+3xh+h²

Which approaches 3x² as h approaches zero.

A = Intersect(xAxis, yAxis)
B = (2.84, 3.62)
C: (0.0, y(B))
D: (x(B), 0.0)
a = Segment(C, B)
b = Segment(D, B)
c = Segment(C, A)
d = Segment(A, D)
h = 0.35 (slider from -0.5 to 0.5, step 0.01)
E: (0.0, y(C) + h)
F: (x(B), y(B) + h)
G: (x(B) + h, y(B) + h)
H: (x(B) + h, y(B))
I: (x(B) + h, 0.0)
J = (-1.42, -1.3)
K: (x(J) + x(C), y(J) + y(C))
L: (x(J) + x(E), y(J) + y(E))
M: (x(J) + x(F), y(J) + y(F))
N: (x(J) + x(B), y(J) + y(B))
O: (x(J) + x(G), y(J) + y(G))
P: (x(J) + x(H), y(J) + y(H))
Q: (x(J) + x(D), y(J) + y(D))
R: (x(J) + x(I), y(J) + y(I))
e = Segment(K, N)
f = Segment(N, B)
g = Segment(C, K)
i = Segment(K, J)
j = Segment(J, Q)
k = Segment(Q, N)
l = Segment(Q, D)
m = Segment(L, K)
n = Segment(E, C)
p = Segment(F, B)
q = Segment(F, E)
r = Segment(E, L)
s = Segment(L, M)
t = Segment(M, N)
a_1 = Segment(M, F)
b_1 = Segment(B, H)
c_1 = Segment(H, I)
d_1 = Segment(I, D)
e_1 = Segment(N, P)
f_1 = Segment(P, R)
g_1 = Segment(Q, R)
h_1 = Segment(I, R)
i_1 = Segment(H, P)
poly1, l_1, m_1, f_2, e_2 = Polygon(L, M, F, E)
S: (h * x(J) / sqrt(x(J)^(2.0) + y(J)^(2.0)), h * y(J) / sqrt(x(J)^(2.0) + y(J)^(2.0)))
T: x(J)^(2.0) + y(J)^(2.0)
U: (x(J) + x(S), y(J) + y(S))
V: (x(K) + x(S), y(K) + y(S))
W: (x(N) + x(S), y(N) + y(S))
Z: (x(Q) + x(S), y(Q) + y(S))
poly2, v, w, z, u = Polygon(V, W, Z, U)
j_1 = Segment(K, V)
k_1 = Segment(N, W)
n_1 = Segment(Q, Z)
p_1 = Segment(J, U)
poly3, k_2, v_1, w_1, n_2 = Polygon(K, V, W, N)
poly4, p_2, h_2, i_2, r_1 = Polygon(P, H, I, R)