Find the average rate of change for f(x)= 1-3x^2 between x=2 and x=2+h

Answer 1

We have the average rate of change is given by [ f( x = 2+ h )m - f(x=2) ] / h and

f ( x=2+h) = 1 - 3(2+h)² = 3( 4 + 4h + h²) and

f(x=2) = 1 - 3(4) So

So [ f(x=2+h) - f(x=2) ] = 12h + 3h² and

[ f(x=2+h) - f(x=2) ] / h = 12 + 3h

Answer 2

The average rate of change is a fancy word for slope.

The slope equation is [f(a)-f(b)]/(a-b)

In this case, a = 2, and b = 2+h

We need to find f(2) and f(2+h):

f(2) = 1-3(2)^2

f(2) = 1-3(4)

f(2) = 1 -12

f(2) = -11

f(2+h) = 1-3(2+h)^2

f(2+h) = 1-3(4+4h+h^2)

f(2+h) = 1-12-12h-3h^2

f(2+h) = -11 -12h-3h^2

So the slop equation is: [-11-(-11 -12h-3h^2)]/[2-(2+h)]

(-11+11+12h+3h^2)/(2-2-h) distribute the negatives

(12h+3h^2)/(-h) combine like terms

[h(12+3h)]/-h factor the numerator

-12-3h cancel common factors