Raymond B. answered • 10/10/21
Math, microeconomics or criminal justice
f(x) = 4x^2 -x + 3
f(a) = 4a^2 -a +3
f(a+h) = 4(a+h)^2 - (a+h) + 3
= 4(a^2 +2ah + h^2) -a -h +3
= 4a^2 + 8ah + 4h^2 -a -h + 3
f(a+h) - f(a) = 8ah +4h^2 -h
[f(a+h) -f(a)]/h = 8a -4h -1
the purpose of this is to explain and calculate a derivative
f'(x)=8x -1= the limit of the difference quotient as h approaches 0
the limit of [f(a+h) - f(a)]/h as h approaches zero
= 8a-4(0)-1
=8a-1
f'(a) is the derivative of f(a) = 8a -1
graphically the derivative is the slope of a tangent line to the curve at the point (a,f(a))
or it's the slope of a secant line which connects the points
(a, f(a)) and (a+h, f(a+h)) as h approaches zero. As h approaches zero, the secant line has the same slope as the tangent line to the curve at the point (a, f(a))
