CHALLENGE: Please answer the question below the problem:

Given f(x) = .04(8x-x2) find the rate of change of elevation when x = 2 without taking a derivative directly. Instead use the slope between f(2-x) and f(2+x).

Solution when x = .5

Substitute .5 in for x: f(2+.5) and f(2-.5)

f(2+.5) = 2.5 and f(2-.5) = 1.5 Substitute each solution into the equation.

f (2.5) = .04(8*2.5 - 2.52)

= .04(20 - 6.25)

= .04 (13.75)

= .55

f (1.5) = .04 (8*1.5 - 1.5^2)

= .04 (12 - 2.25)

= .04 (9.75)

= .39

To find Rate of Change, calculate {f (2.5) - f (1.5)} / {2.5 - 1.5}

f(2.5) = .55

f(1.2) = .39

2.5 - 1.5 = 1

So, .55 - .39= .16

.16 / 1 = .16

The rate of change, or slope, is 0.16

QUESTION: Many students seem to be getting very close to the same answer. Why is that true? Would it be true for any function?