I already know that the solution is correct please answer the questions at the end of the equation.

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).

If x = .5

f (2+.5) =2.5

f (2 -.5) =1.5

f (2.5) = .04 (8*2.5 - 2.5²)

= .04(20 - 6.25)

= .04 (13.75)

=.55

f (1.5) = .04 (8*1.5 - 1.5²)

= .04 (12 - 2.25)

= .04 (9.75)

=.39

Find the rate of change

f (2.5) - f (1.5) / (2.5 - 1.5)

f (2.5) = .55

f (1.5) = .39

2.5-1.5= 1

So,.55-.39= .16

.16 / 1 = .16

The rate of change, or slope, is 0.16

Students are getting the same answer when substituting:

x = .1

x = .2

x = .3

x = .4

x = .5

x = .6

x = .7

x = .8

x = .9

Question:

1) Why is that true?

2) Would it be true for any function?