4.2 Linear Approximations and Differentials

We have just seen how derivatives allow us to compare related quantities that are changing over time. In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher-degree polynomials Introduction to Power Series and Functions.

Linear Approximation of a Function at a Point

Consider a function $f$ that is differentiable at a point $x = a .$ Recall that the tangent line to the graph of $f$ at $a$ is given by the equation

y = f (a) + f^{'} (a) (x - a) .

For example, consider the function $f (x) = \frac{1}{x}$ at $a = 2.$ Since $f$ is differentiable at $x = 2$ and $f^{'} (x) = - \frac{1}{x ^{2}},$ we see that Therefore, the tangent line to the graph of at is given by the equation

y = \frac{1}{2} - \frac{1}{4} (x - 2) .

Figure 4.7(a) shows a graph of $f (x) = \frac{1}{x}$ along with the tangent line to $f$ at $x = 2.$ Note that for $x$ near 2, the graph of the tangent line is close to the graph of $f .$ As a result, we can use the equation of the tangent line to approximate for near 2. For example, if the value of the corresponding point on the tangent line is

y = \frac{1}{2} - \frac{1}{4} (2.1 - 2) = 0.475.

The actual value of $f (2.1)$ is given by

f (2.1) = \frac{1}{2.1} \approx 0.47619.

Therefore, the tangent line gives us a fairly good approximation of $f (2.1)$ (Figure 4.7(b)). However, note that for values of $x$ far from 2, the equation of the tangent line does not give us a good approximation. For example, if $x = 10,$ the $y$ -value of the corresponding point on the tangent line is

y = \frac{1}{2} - \frac{1}{4} (10 - 2) = \frac{1}{2} - 2 = - 1.5,

whereas the value of the function at $x = 10$ is $f (10) = 0.1.$

This figure has two parts a and b. In figure a, the line f(x) = 1/x is shown with its tangent line at x = 2. In figure b, the area near the tangent point is blown up to show how good of an approximation the tangent is near x = 2. — Figure 4.7(a) The tangent line to $f (x) = 1 / x$ at $x = 2$ provides a good approximation to $f$ for $x$ near 2. (b) At $x = 2.1,$ the value of $y$ on the tangent line to is 0.475. The actual value of is which is approximately 0.47619.

In general, for a differentiable function $f,$ the equation of the tangent line to $f$ at $x = a$ can be used to approximate $f (x)$ for $x$ near $a .$ Therefore, we can write

f (x) \approx f (a) + f^{'} (a) (x - a) for x near a .

We call the linear function

L (x) = f (a) + f^{'} (a) (x - a)

the linear approximation, or tangent line approximation, of $f$ at $x = a .$ This function $L$ is also known as the linearization of $f$ at $x = a .$

To show how useful the linear approximation can be, we look at how to find the linear approximation for $f (x) = x$ at $x = 9.$

Linear approximations may be used in estimating roots and powers. In the next example, we find the linear approximation for $f (x) = (1 + x)^{n}$ at $x = 0,$ which can be used to estimate roots and powers for real numbers near 1. The same idea can be extended to a function of the form $f (x) = (m + x)^{n}$ to estimate roots and powers near a different number

Differentials

We have seen that linear approximations can be used to estimate function values. They can also be used to estimate the amount a function value changes as a result of a small change in the input. To discuss this more formally, we define a related concept: differentials. Differentials provide us with a way of estimating the amount a function changes as a result of a small change in input values.

When we first looked at derivatives, we used the Leibniz notation $d y / d x$ to represent the derivative of $y$ with respect to $x .$ Although we used the expressions dy and dx in this notation, they did not have meaning on their own. Here we see a meaning to the expressions dy and dx. Suppose $y = f (x)$ is a differentiable function. Let dx be an independent variable that can be assigned any nonzero real number, and define the dependent variable $d y$ by

d y = f^{'} (x) d x .

It is important to notice that $d y$ is a function of both $x$ and $d x .$ The expressions dy and dx are called differentials. We can divide both sides of Linear Approximations and Differentials by $d x,$ which yields

\frac{d y}{d x} = f^{'} (x) .

This is the familiar expression we have used to denote a derivative. Linear Approximations and Differentials is known as the differential form of Linear Approximations and Differentials.

We now connect differentials to linear approximations. Differentials can be used to estimate the change in the value of a function resulting from a small change in input values. Consider a function $f$ that is differentiable at point $a .$ Suppose the input $x$ changes by a small amount. We are interested in how much the output $y$ changes. If $x$ changes from $a$ to $a + d x,$ then the change in $x$ is $d x$ (also denoted and the change in is given by

Δ y = f (a + d x) - f (a) .

Instead of calculating the exact change in $y,$ however, it is often easier to approximate the change in $y$ by using a linear approximation. For $x$ near $a,$ $f (x)$ can be approximated by the linear approximation

L (x) = f (a) + f^{'} (a) (x - a) .

Therefore, if $d x$ is small,

f (a + d x) \approx L (a + d x) = f (a) + f^{'} (a) (a + d x - a) .

That is,

f (a + d x) - f (a) \approx L (a + d x) - f (a) = f^{'} (a) d x .

In other words, the actual change in the function $f$ if $x$ increases from $a$ to $a + d x$ is approximately the difference between $L (a + d x)$ and $f (a),$ where $L (x)$ is the linear approximation of $f$ at By definition of this difference is equal to In summary,

Δ y = f (a + d x) - f (a) \approx L (a + d x) - f (a) = f^{'} (a) d x = d y .

Therefore, we can use the differential $d y = f^{'} (a) d x$ to approximate the change in $y$ if $x$ increases from $x = a$ to $x = a + d x .$ We can see this in the following graph.

A function y = f(x) is shown along with its tangent line at (a, f(a)). The tangent line is denoted L(x). The x axis is marked with a and a + dx, with a dashed line showing the distance between a and a + dx as dx. The points (a + dx, f(a + dx)) and (a + dx, L(a + dx)) are marked on the curves for y = f(x) and y = L(x), respectively. The distance between f(a) and L(a + dx) is marked as dy = f’(a) dx, and the distance between f(a) and f(a + dx) is marked as Δy = f(a + dx) – f(a). — Figure 4.11The differential $d y = f^{'} (a) d x$ is used to approximate the actual change in $y$ if $x$ increases from $a$ to $a + d x .$

We now take a look at how to use differentials to approximate the change in the value of the function that results from a small change in the value of the input. Note the calculation with differentials is much simpler than calculating actual values of functions and the result is very close to what we would obtain with the more exact calculation.

Calculating the Amount of Error

Any type of measurement is prone to a certain amount of error. In many applications, certain quantities are calculated based on measurements. For example, the area of a circle is calculated by measuring the radius of the circle. An error in the measurement of the radius leads to an error in the computed value of the area. Here we examine this type of error and study how differentials can be used to estimate the error.

Consider a function $f$ with an input that is a measured quantity. Suppose the exact value of the measured quantity is $a,$ but the measured value is $a + d x .$ We say the measurement error is dx (or $Δ x) .$ As a result, an error occurs in the calculated quantity $f (x) .$ This type of error is known as a propagated error and is given by

Δ y = f (a + d x) - f (a) .

Since all measurements are prone to some degree of error, we do not know the exact value of a measured quantity, so we cannot calculate the propagated error exactly. However, given an estimate of the accuracy of a measurement, we can use differentials to approximate the propagated error $Δ y .$ Specifically, if $f$ is a differentiable function at $a,$ the propagated error is

Δ y \approx d y = f^{'} (a) d x .

Unfortunately, we do not know the exact value $a .$ However, we can use the measured value $a + d x,$ and estimate

Δ y \approx d y \approx f^{'} (a + d x) d x .

In the next example, we look at how differentials can be used to estimate the error in calculating the volume of a box if we assume the measurement of the side length is made with a certain amount of accuracy.

The measurement error dx $(=Δ x)$ and the propagated error $Δ y$ are absolute errors. We are typically interested in the size of an error relative to the size of the quantity being measured or calculated. Given an absolute error $Δ q$ for a particular quantity, we define the relative error as $\frac{Δ q}{q},$ where $q$ is the actual value of the quantity. The is the relative error expressed as a percentage. For example, if we measure the height of a ladder to be 63 in. when the actual height is 62 in., the absolute error is 1 in. but the relative error is or By comparison, if we measure the width of a piece of cardboard to be 8.25 in. when the actual width is 8 in., our absolute error is in., whereas the relative error is or Therefore, the percentage error in the measurement of the cardboard is larger, even though 0.25 in. is less than 1 in.

Key Concepts

A differentiable function $y = f (x)$ can be approximated at $a$ by the linear function
$L (x) = f (a) + f^{'} (a) (x - a) .$

Key Equations

Linear approximation	$L (x) = f (a) + f^{'} (a) (x - a)$
A differential	$d y = f^{'} (x) d x .$