Local Linear Approximation
The concept of Local Linearity allows us to approximate the value of a function near a point using the tangent line at that point. This is called the local linear approximation.
Idea:
If we zoom in sufficiently close to a point on the graph of a function, the graph starts to look more and more like a straight line. This straight line is the tangent line.
Formula:
The local linear approximation of $ f(x) $ at $ x=a $ is given by:
$$ f(x) \approx f(a) + f’(a)(x-a) $$
Example:
Let’s approximate the value of $ \sqrt{4.1} $ using the local linear approximation of $ f(x) = \sqrt{x} $ at $ x=4 $ .
We have $ f’(x) = \frac{1}{2\sqrt{x}} $ , so $ f’(4) = \frac{1}{4} $ .
Using the local linear approximation, we get:
$$ \sqrt{4.1} \approx f(4) + f’(4)(4.1-4) = 2 + \frac{1}{4}(0.1) = 2.025 $$
This approximation is quite accurate, as the actual value of $ \sqrt{4.1} $ is approximately 2.0248.