We use a method called “linear approximation” to estimate the value of a (complicated) function at a given point.

*linear approximation*is a fancy phrase for something you already know:

**The line tangent to the graph of a function at a point is
very close to the graph of the function near that point.**

This tangent line is the graph of a linear function, called the **linear approximation**.

at the point where are given in the figure below. Find the equation of the tangent line.

Now, we define a function, , by . This function is linear and its graph is the line tangent to the curve at the point where . This function deserves a special name.

Note that the graph of is just the tangent line to the graph of at .

A linear approximation of is a “good” approximation as long as is “not too far” from . If one “zooms in” on the graph of sufficiently, then the graphs of and are nearly indistinguishable.

As a first example, we will see how linear approximations allow us to approximate “difficult” computations.

What would happen if we chose instead?

Then we would use , the linear approximation to the function at . In that case, . The graph of , together with the graphs of and is given in the figure below.

.

So, our choice, , was better!

### Differentials

The graph of a function and the graph of , the linear approximation of at , are shown in the figure below. Also, two quantities, and , and a point are marked in the figure. Look carefully at the figure when answering the questions below.

So, we can write and call it a **differential** of at . Notice that we can define a
differential at any point of the domain of , provided that exists. We will do that in
our next definition.

**differential of**. We define , a differential of , at a point by

Geometrically, differentials can be interpreted via the diagram below.

Note, it is now the case (by definition!) that

Essentially, differentials allow us to solve the problems presented in the previous examples from a slightly different point of view. Recall, when is near but not equal zero, Hence, We can replace a quantity with a quantity to write

Adding to both sides we see or, equivalently There are contexts where the language of differentials is common. Here is the basic strategy:

We will repeat our previous examples using differentials.

The upshot is that linear approximations and differentials are simply two slightly different ways of doing the exact same thing.

### Error approximation

Differentials also help us estimate error in real life settings.

There is a mark indicating when the glass is filled to ml at cm from the base of the glass. If the glass is filled within millimeters of the mark, what are the bounds on the volume?

### New and old friends

You might be wondering, given a plot ,

What’s the difference between and ? What about and ?

Regardless, it is now a pressing question. Here’s the deal: is the **average rate of
change** of with respect to . On the other hand: is the **instantaneous rate of
change** of with respect to . Essentially, and are the same type of thing, they
are (usually small) changes in . However, and are very different things.

- ; it is the change in associated to .
- , it is the change in associated to . Note: .
So, the change