1. §3.10 Linear Approximations and Differentials
Main idea
We have seen that a curve lies very close to its
tangent line near the point of tangency. So we can
use the tangent line to approximate the graph near
the point.
Topics:
Linear approximation
Differentials

2. I. linear approximation
The tangent line at (a, f(a)) is
y = f(a) + f (a)(x – a)
The approximation
f(x)  f(a) + f (a)(x – a)
is called linear
approximation of f at a.
The linear function
L(x) = f(a) + f (a)(x – a)
is called the linearization of f at a.

6. Differentials
Consider y = f(x). Then we have
We treat the left side as a ratio of two quantities.
Then we have

7. Definition: In equation (2) on last slide, both dx
and dy are called differentials. dy is the dependent
variable and dx is the independent variable. dy
depends on both dx and x. So dy is a two variable
function.
Comment: Computing derivatives y' = f ' (x) is
equivalent to computing differentials dy = f ' (x)dx .

8. Increment and differential
If x is given an increment (i.e. change x), then
y = f(x + x) – f(x) is the increment of y.
The differential dy can be used to approximate
y when dx = x is small. That is:
y  dy
f(x + x) – f(x)  f (x)x.
Note: Computing y is hard
in general. But computing dy
is always easy.

9. Relationship between linear approximation
and differential:
L(x) = f(a) + f (a)(x – a) = f(a) + dy
where dy = f (a) dx and dx = x = x – a.
L(x) – f(a) = dy
f(x)  L(x) = f(a) + dy
 f(x) – f(a)  dy  y  dy
where y = f(x) – f(a)