1. Differentials and Linear Approximation

2. Differentials dy = 𝑑𝑥 ∆𝑦 ∆𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥
When we first started to talk about derivatives, we said that becomes when ∆x and ∆y become very small.
dy can be considered a very small change in y.
dx can be considered a very small change in x.
Let be a differentiable function.
dy = 𝑑𝑥
𝑑𝑦 𝑑𝑥

3. Linear approximation is a very easy thing to do, and once you master it, you can impress all of your friends by calculating things like in your head about 4.125! Impressed?
I’ll teach you how.
Recall that if a function 𝑓 (𝑥) is differentiable at 𝑥=𝑎 , we say it is locally linear at 𝑥=𝑎 . This means that as we zoom in closer and closer and closer and closer around 𝑥=𝑎 , the graph of 𝑓 (𝑥) , regardless of how curvy it is, will begin to look more and more and more and more like the tangent line at 𝑥=a .

4. This means that we can use the equation of the tangent line of 𝑓 (𝑥) at 𝑥=𝑎 to approximate 𝑓 (𝑥) for values close to 𝑥=𝑎. Let’s take a look at and the figure below.
𝑓 𝑥 = 𝑥 1 3
𝑓 ′ 𝑥 = 𝑥 − 2 3
𝑓 ′ 64 = − 2 3 = 1 48
tangent line at (64,4)
𝑦= (𝑥−64)
𝑓 70 ≈𝑦 70 = −64 = =4.125

5. Linear approximation of 𝑓(𝑥) at point 𝑥 around 𝑎
Tangent line approximation
Linearization of f around a :

6. Linear approximation of 𝑓(𝑥) at point 𝑥 around 𝑎
1. step: Assuming that 𝑓(𝑥) is differentiable at 𝑥 = 𝑎, from the picture:
𝑓 𝑥 = 𝑓 𝑎 + ∆𝑓 ≈ 𝑓(𝑎) + 𝑓′(𝑎)∆𝑥
Linear approximation of 𝑓(𝑥) at point 𝑥 around 𝑎
𝑓(𝑥) =𝑓(𝑎) + 𝑓′(𝑎)(𝑥−𝑎)
𝐹𝑜𝑟 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 𝑥 𝑐𝑙𝑜𝑠𝑒 𝑡𝑜 𝑎, 𝑙𝑖𝑛𝑒𝑎𝑟 𝑎𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓 𝑥 𝑖𝑠
𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑦 𝑥 𝑎𝑡 𝑡ℎ𝑎𝑡 𝑝𝑜𝑖𝑛𝑡 𝑥
𝑓 𝑥 =𝑦(𝑥)
𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 𝑹 is small for the points close to point 𝑎

7. STEPS for linear approximation of a function 𝑓(𝑥) for points 𝑥 around 𝑎:
Find the equation of the tangent line 𝑦 𝑥 at the center 𝑎,𝑓(𝑎) in point-slope form.
𝑦(𝑥)=𝑓 𝑎 +𝑓′(𝑎) 𝑥−𝑎)
2. For some 𝑥=𝑐 around 𝑎 : 𝑓 𝑐 ≈𝑦(𝑐)
3. If asked, determine if 𝑦(𝑐) is an over-approximation or an under approximation by examining the concavity of 𝑓 (𝑥) at the center 𝑥=𝑎
a. If 𝑓′′ 𝑎 <0, 𝑓 (𝑥) is concave down around 𝑎, tangent line
is above the curve and 𝑦(𝑐) is an over-approximation
b. If 𝑓 ′′ 𝑎 >0, 𝑓 (𝑥) is concave up around 𝑎 , tangent line
is bellow the curve and 𝑦(𝑐) is an under-approximation

8. Are these approximations overestimates or underestimates?
example:
Find the linearization of the function 𝑓 𝑥 = 𝑥+3 at 𝑎=1, and use it to approximate the numbers and
Are these approximations overestimates or underestimates?
𝑓 ′ 𝑥 = 1 2 𝑥+3
𝑓 𝑥 ≈𝑓 𝑓 ′ 𝑥 𝑥=1 (𝑥−1)
𝑓 𝑥 ≈ 𝑥−1 = 𝑥 4
linear aproximation: 𝑥+3 ≈ 𝑥 (when 𝑥 is near 1)
3.98 ≈ =1.995
𝑓 ′′ 1 <0 → 𝑓 𝑥 is concave down around 𝑥=1
Tangent line is above the curve,
so estimated values are overestimate
4.05 ≈ =2.0125