Linear Approximation Equation

Linear Approximation: 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.

This lesson shows how to find a linearization of a function and how to use it to make a linear approximation. This method is used quite often in many fields of science, and it requires knowing a bit about calculus, specifically, how to find a derivative.

Linear Approximation Equation

Linear Approximation Calculator

In this section we’re going to take a look at an application not of derivatives but of the tangent line to a function. Of course, to get the tangent line we do need to take derivatives, so in some way this is an application of derivatives as well.

Take a look at the following graph of a function and its tangent line.

This is a graph of an unknown function that looks like the right side of an upwards opening parabola whose vertex is on the y-axis. Also shown on the graph is the tangent line to this graph at the point (a, f(a)). The tangent line falls below the graph of the function.

From this graph we can see that near \(x = a\) the tangent line and the function have nearly the same graph. On occasion we will use the tangent line, \(L\left( x \right)\), as an approximation to the function, \(f\left( x \right)\), near \(x = a\). In these cases we call the tangent line the linear approximation to the function at \(x = a\).

Example. Consider the function y = f(x) = 5x2. Let $\Delta x$ be an increment of x. Then, if $\Delta y$ is the resulting increment of y, we have

\begin{displaymath}\begin{array}{lll} \Delta y &=& f(x+ \Delta x) - f(x)\\ &=&... ...\ &=& 10 x (\Delta x) + 5 (\Delta x )^2\;\cdot\\ \end{array}\end{displaymath}

On the other hand, we obtain for the differential dy:

\begin{displaymath}dy = f'(x)\,dx = 10 x\,dx\;.\end{displaymath}

In this example we are lucky in that we are able to compute $\Delta y$ exactly, but in general this might be impossible. The error in the approximation, the difference between dy(replacing dx by $\Delta x$) and $\Delta y$, is $5 (\Delta x )^2$, which is small compared to $\Delta x$.

Linear approximation is a method of estimating the value of a function, f(x), near a point, x = a, using the following formula: The formula we’re looking at is known as the linearization of f at x = a, but this formula is identical to the equation of the tangent line to f at x = a.

Linear approximation, or linearization, is a method we can use to approximate the value of a function at a particular point. The reason liner approximation is useful is because it can be difficult to find the value of a function at a particular point. Square roots are a great example of this.

In calculus, the terms linear approximationlinearization, and tangent line approximation all refer to the same thing. There are other linear approximations used in mathematics besides this one. For instance, in statistics, regression analysis is used to fit the “best” linear function to a set of data.

Linear Approximation Formula

So, how do you find the linearization of a function f at a point x = a? Remember that the equation of a line can be determined if you know two things:

  1. The slope of the line, m
  2. Any single point that the line goes through, (ab).

We plug these pieces of info into the point-slope form, and this gives us the equation of the line. (This is just algebra, folks; no calculus yet.)

y – b = m(xa)

But, in problems like these, you will not be given values for b or m. Instead, you have to find them yourself. Firstly m = ‘(a), because the derivative measures the slope, and secondly, b = f(a), because the original function measures y-values.

Let x0 be in the domain of the function f(x). The equation of the tangent line to the graph of f(x) at the point (x0,y0), where y0 = f(x0), is

\begin{displaymath}y - y_0 = f'(x_0) (x-x_0) \cdot\end{displaymath}

If x1 is close to x0, we will write $x_1 = x_0 + \Delta x$, and we will approximate $f(x_0 + \Delta x)$ by the point (x1,y1) on the tangent line given by

\begin{displaymath}y_1 = y_0 + \Delta x f'(x_0) \cdot\end{displaymath}

If we write $\Delta y = y_1 - y_0$, we have

\begin{displaymath}\Delta y = \Delta x f'(x_0) \cdot\end{displaymath}

In fact, one way to remember this formula is to write f‘(x) as $\displaystyle \frac{dy}{dx}$ and then replace d by $\Delta$. Recall that, when x is close to x0, we have

\begin{displaymath}f(x) \approx f(x_0) + f'(x_0)(x-x_0) \;\cdot\end{displaymath}

Local Linear Approximation

Given a twice continuously differentiable function {\displaystyle f} of one real variable, Taylor’s theorem for the case {\displaystyle n=1} states that

{\displaystyle f(x)=f(a)+f'(a)(x-a)+R_{2}\ }

where {\displaystyle R_{2}} is the remainder term. The

linear approximation is obtained by dropping the remainder:

{\displaystyle f(x)\approx f(a)+f'(a)(x-a)}.

This is a good approximation for {\displaystyle x} when it is close enough to {\displaystyle a}; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of {\displaystyle f} at {\displaystyle (a,f(a))}. For this reason, this process is also called the tangent line approximation.

If {\displaystyle f} is concave down in the interval between {\displaystyle x} and {\displaystyle a}, the approximation will be an overestimate (since the derivative is decreasing in that interval). If {\displaystyle f} is concave up, the approximation will be an underestimate.

Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function {\displaystyle f(x,y)} with real values, one can approximate {\displaystyle f(x,y)} for {\displaystyle (x,y)} close to {\displaystyle (a,b)} by the formula

The right-hand side is the equation of the plane tangent to the graph of {\displaystyle z=f(x,y)} at {\displaystyle (a,b).}

In the more general case of Banach spaces, one has

{\displaystyle f(x)\approx f(a)+Df(a)(x-a)}

where {\displaystyle Df(a)} is the Fréchet derivative of {\displaystyle f} at {\displaystyle a}.