Thread: The First Derivative and Linear Approximations: What's the Difference

Hello Everyone,

What are linear approximations for? I know the mechanics of how to arrive at the answer using the formulas, but I don't know what the answers mean. Allow me to elaborate.

Wikipedia says:

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value.

and

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

From my own calculus study, this tells me that the first derivative of a function is the slope of that function at some point. That slope is also the tangent of that function at the same point.

A linear approximation gives the formula for the tangent of a function at some point, but it is an "approximation".

What's the difference between the two? Why would I want the linear approximation over the first derivative which is so much less complicated to compute? If it would be okay, could you give an example when I would use one or the other? Thank you.

calculus-user

The difference between the derivative is the same as the difference between the slope of a line and its eqn.

The derivative is the slope
Thelinear approx is the the tangent line itself

derivative -- f '(a)

linear approximation y = f ' (a) (x-a) +f(a)

The derivative is the instaneous rate of change so whenever you want the inst rate of change of a function.

an example of where the linear approximation is used in error analysis once you learn the differential, which is the change in f(x) as you move along the tangent line.

If you want see the notes on the differential and local linearity
at Calculus 1