linear approximation

[Sanguinne]

i understand the linear approximation formula
but i dont understand when questions ask for the approximate change and percentage
can someone give me a brief explanation on them

[brightsky]

consider the following problem.

you are given f(a). you are given f'(a). but you are not given f(x).  (a by the way is some constant in the domain of f(x)). you are now asked to find f(a+h), where h is small.

you are not given f(x), so there is no way you can work out with perfect accuracy f(a+h). you are given f(a) and f'(a). so you can sketch that point. but from there, the graph can go anywhere. you don't know.

so we can only approximate. construct a tangent to the point x = a (we can because we know f(a) and f'(a)). [the equation of the tangent would be given by y - f(a) = f'(a) (x-a).] now we approximate f(a+h) with the y-value of the point on this tangent where x = a+h. since h is small, this is a pretty good approximation, provided that the graph doesn't do something crazy after x = a.

so what is the approximation?

well the equation of the tangent is y - f(a) = f'(a) (x-a). so at x = a+h:

y-f(a) = f'(a)*(a+h-a)
y-f(a) = f'(a)*h
y = f(a) + h*f(a)

since y is an approximation for f(a+h),
f(a+h) approximately equals to f(a) + h*f(a).

advice: pay no attention to the formula. you will have to memorise it at some stage but for now, it is much better for you to understand how linear approximation works geometrically. you are approximating a point on some curve with a point on the tangent.