Imagine you plot a smooth curve, let’s say function \(\boldsymbol{f\left(x\right)}\), on a graphing utility. Pick a point \(\boldsymbol{\left(c , f\left(c\right) \right)}\) and draw a tangent line. Then, zoom in several times. As the area near the point is repeatetively enlarged, the curve becomes more and more like the tangent line. Smooth curves appear straighter on smaller scales and it is the basis of many important mathematical ideas, one of which is linear approximation. The three graphs below show this zooming effect.
The main idea of linear approximation is to use the line tangent at a point \(\boldsymbol{\left(c , f\left(c\right) \right)}\) to approximate the value of the function at other points close to \(\boldsymbol{\left(c , f\left(c\right) \right)}\). This is very useful when the function is a complex one; evaluating nearby points using a linear function is much simpler. I have read that this simplification is used very often in science and engineering.
Here’s how it works: assume that \(\boldsymbol{f}\) is differentiable on an interval containing the point \(\boldsymbol{c}\) and the slope of the tangent line to the curve at the point \(\boldsymbol{c}\) is \(\boldsymbol{f’\left(c\right)}\). Then, the equation of the tangent line is:
\[y\; – f\left(c\right) = f'(c) \left(x – c\right)\quad \text{ or }\quad y = f\left(c\right) + f'(c) \left(x\, – c\right)\]
This tangent line represents a new function \(\boldsymbol{L}\) that we call the linear approximation to \(\boldsymbol{f}\) at the point \(\boldsymbol{c}\). That is, function values near \(\boldsymbol{c}\) can be easily approximated using the linear approximation \(\boldsymbol{L}\):
\[f\left(x\right) \approx L\left(x\right) = f\left(c\right) + f'(c)\left(x\, – c\right)\]
Example One
Find the linear approximation to \(\boldsymbol{f\left(x\right) = sin\left(x\right) \text{ at } x = 0 \text{ and use it to approximate } sin\left(2.5^{o}\right)}\).
\begin{align*}
\text{First, we make some definitions:}\, f\left(x\right) &= sin\left(x\right),\quad f'(x) = cos\left(x\right),\quad c = 0 \\
\text{Construct the linear approximation:}\, L\left(x\right) &= f\left(c\right) + f'(c)\left(x\, – c\right) \\
&= sin\left(0\right) + cos\left(0\right)\left(x\, – 0\right) \\
&= 0 + 1\left(x\, – 0\right) \\
&= x \\
\text{So, the linear approximation to} \\
\text{the } sin \text { function at } x = 0 \text { is: } L\left(x\right) &= x\qquad \text{Simple, isn’t it?}
\end{align*}
Now, let’s use it to approximate \(\boldsymbol{sin\left( 2.5^{o} \right)}\), but first, we need to convert from degrees to radians: \(\boldsymbol{2.5^{o}\left(\frac{\pi}{180}\right) = 0.04363\, rad.}\).
Let’s see what we get:
\begin{align*}
sin\left(2.5^{o}\right) \approx L\left(0.04363\right) &= 0.04363 \\
\text{A graphing utility gives:}\, sin\left(2.5^{o}\right) &\approx 0.04362 \\
\text{The approximation is accurate to four decimal places.}
\end{align*}
Example 2
Approximate the change in the volume of a sphere when its radius changes from \(\boldsymbol{r = 5\:}\) to \(\boldsymbol{\:r = 5.1}\).
\begin{align*}
\text{First, we make some definitions:}\; V\left(r\right) &= \frac{4}{3}\pi\:r^{3},\quad V'(r) = 4\pi\:r^{2},\quad c = 5 \\
\text{Construct the linear approximation:}\; L\left(r\right) &= V\left(c\right) + V'(c)\left(r\, – c\right) \\
&= \frac{4}{3}\pi\:\left(5\right)^{3} + 4\pi\:\left(5\right)^{2}\left(r\, – 5\right) \\
&= \frac{500 \pi}{3} + 100 \pi \left(r\, – 5\right) \\
\text{We can use it for any value near } c = 5:\, L\left(r\right) &= 100 \pi\:r\; – \frac{1000 \pi}{3}
\end{align*}
\begin{align*}
\text{We are asked about the change in the volume.} \\
\text{Let’s see what we get:}\, L\left(5.1\right)\, – L\left(5\right) &\approx 31.416 \\
V\left(5.1\right)\, – V\left(5\right) &\approx 32.048 \\
\text{The approximation is off by about 1.97%.}
\end{align*}
In the next blogpost I will write about another technique usually studied at the same time as Linear Approximation; it is called Differentials.
If you find any errors in this post, or have some thoughts about it, please, email me at [email protected] .