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Question

suppose that f(x)=3x-1. find f(a+h)-f(a)/h

Answer 1

For f(x) = 3x-1, {f(a+h) - f(a)}/h is written as [3(a+h)-1]-[3a-1]/h which simplifies to

[3a+3h-1-3a+1]/h or 3h/h or 3.

Note that for f(x) = 3x-1, which is the equation of a straight line, the slope of the line m

(from the slope-intercept form y=mx+b) is the coefficient of 3x equal to 3.

This slope is also obtained by taking the limit of [3(a+h)-1]-[3a-1]/h above as h goes to 0, or

lim_(h→0)[3(a+h)-1]-[3a-1]/h, again equal to 3.