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