1. 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

2. 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

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