On a small neighborhood The function is approximately linear Lesson: ____ Section: 3.09 Local Linearity & Tangent Line Approximation Warmup: Ex. Write the equation of the line through (5,8) with a slope of 2. Ex. Write the equation of the tangent line to 𝒇 𝒙 𝐚𝐭 𝒙=𝒂. If we sufficiently zoom in on any differentiable function, it will look like a straight line. We say that differentiable functions have … a f(a) “local linearity” On a small neighborhood The function is approximately linear

Note that error is a function So, for values near a, the function resembles the tangent line to the curve at a. 𝒇 𝒙 ≈𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒍𝒊𝒏𝒆 𝒂𝒕 𝒂 This is called the “local linearization” of function f near a. 𝒇 𝒙 ≈𝒇 𝒂 +𝒇′(𝒂)(𝒙−𝒂) By plugging into this simple linear equation, we can find the approximate location of other points on the original function. This technique is called tangent line or linear approximation The Error, or E(x), in this approximation is defined by: Error = Actual - Estimate 𝑬 𝒙 =𝒇 𝒙 −[𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒍𝒊𝒏𝒆 𝒂𝒑𝒑𝒓𝒐𝒙𝒊𝒎𝒂𝒕𝒊𝒐𝒏] 𝑬 𝒙 =𝒇 𝒙 −[𝒇 𝒂 +𝒇′(𝒂)(𝒙−𝒂)] Note that error is a function

Ex. a. Find the local linearization for 𝑓 𝑥 = 𝑥 near x = 1. b. Use this to approximate f (1.1)