Linear Approximation

The primary formula we use for linear approximations:

The formula above states that the value of some function ${f,}$ at a given input ${x}$ is approximately the sum of ${f(x_0)}$ and ${f'(x_0) \times (x - x_0),}$ where ${x_0}$ is a base point. A level deeper, the formula provides that if we were given some curve ${f,}$ its value at ${x}$ is approximately the same as the value ${x}$ at its tangent line.

For example, consider ${f(x) = \ln x.}$ We know ${f'(x)= \dfrac{1}{x}.}$ Let's pick a base point, say, ${x_0 = 1.}$ We use ${1}$ because it's a point where we immediately know the value for the natural logarithm ${\ln 1 = 0.}$ Where ${x_0 = 1,}$ we have:

• ${f(1) = \ln 1 = 0}$
• ${f'(1) = \dfrac{1}{1} = 1}$

Applying the approximation formula to our facts:

Thus, we have:

If we plot ${f(x) = \ln x}$ and ${g(x) = x - 1,}$ we see the following:

What the graph above tells us is that the value of ${\ln x}$ is roughly ${x - 1.}$ The key word, however, is “roughly.” This is just an approximation. And with our plot, we can see that the approximation is good as long as we're somewhere near ${x = 1.}$ As we venture further and further from ${x = 1,}$ the less accurate the approximation is.

Another way to interpret this approximation method is to refer to the definition of a derivative. Recall this definition:

Reading this expression from the other direction, we have:

This effectively states that the average rate of change, when ${\Delta x}$ is very near zero, is the derivative. This means that:

In other words, the average rate of change is approximately the infinitesimal rate of change. All together, the two formulas below are essentially the same:

1. ${f(x) \approx f(x_0) + [f'(x_0) \cdot (x - x_0)]}$
2. ${\dfrac{\Delta f}{\Delta x} \approx f'(x_0)}$

To see how, we manipulate the second formula. Multiplying ${\Delta x}$ to both sides:

Evaluating ${\Delta f}$ and ${\Delta x:}$

Adding ${f(x_0)}$ to both sides, we get the first formula:

Thus, formulas (1) and (2) are simply two sides of the same coin, with each providing a unique way of interpreting approximations.

When we work with linear approximations, we always want to have the base point ${x_0 = 0.}$ Why? Because it result in a more readable formula:

That said, there are a few precautions to keep in mind with our formulas. The first formula,

only works when ${x \approx x_0.}$ The second formula,

only works when ${x \approx 0.}$ With these rules in mind, let's consider a few key functions: ${y = \sin x,}$ ${y = \cos x,}$ and ${y = e^x.}$

For ${y = \sin x,}$ we know that ${y' = \cos x.}$ Laying the values out in a table:

Applying our formula to our findings:

Hence, we can infer:

Next, the function ${y = \cos x.}$ We know that ${y' = - \sin x.}$ The table:

Thus, we have:

Accordingly, we conclude:

The same analysis extends to ${f(x) = e^x.}$

Once more this yields:

Let's consider two more useful approximations. These the approximations for:

1. ${f(x) = \ln(1 + x)}$
2. ${f(x) = (1 + x)^r}$

Examining ${f(x) = \ln(1+x),}$ we have:

And for ${f(x) = (1+x)^r,}$ we have:

Our analysis yields the following conclusions:

1. ${\large \ln(1+x) \approx x}$
2. ${\large (1+x)^r \approx 1 + rx}$

Why are these approximations useful? Because they allow us to think about function outputs in a much simpler way. For example, what's the value of ${\ln(1.1)?}$ Most of us don't know off the top of our heads. But if we knew about linear approximation, we'd find that:

Here's another example:

Problem: Rational-natural-radicand. Find a linear approximation near ${x=0}$ (${x \approx 0}$) of the function:

$$\dfrac{e^{-3x}}{\sqrt{1+x}}$$

The first step to this problem is to rewrite in such a way that we can apply our linear approximation formulas. In this case:

Applying our formulas:

If we carry out the multiplication:

With the expanded expression, it turns out that we can throw out the ${\dfrac{3}{2}x^2}$ term. Why? Because when we performed the original linear approximation, we threw out quite a few terms. In fact, the expanded expression above omits many other terms. Because we took that liberty in our original approximation, we can take that liberty again here:

Hence:

That's very cool. We've effectively come up with a way to reason about the original, nasty rational function through a much simpler interpretation.