Figure

EXAMPLE 2Finding the Linear Approximation to a Function

Find the linear approximation $L( x)$ to $f(x)=\sin x$ near $x=0$ .
Use $L( x)$ to approximate $\sin \left( -0.3\right)$ , $\sin 0.1$ , $\sin 0.4$ , $\sin 0.5$ , and $\sin \dfrac{\pi }{4}$ .
Graph $f$ and $L.$

(a) Since $f^\prime ( x) =\cos x$ , then $f(0)=\sin 0=0$ and $f^\prime (0) = \cos 0 = 1$ . Using Equation (3), the linear approximation $L( x)$ to $f$ at $0$ is $L( x) =f(0) + f^\prime (0)(x -0)=x$

So, for $x$ close to $0,$ the function $f( x) =\sin x$ can be approximated by the line $L( x) =x.$

(b) The approximate values of $\sin x$ using $L( x) =x,$ the true values of $\sin x,$ and the absolute error in using the approximation are given in Table 3. From Table 3, we see that the further $x$ is from $0,$ the worse the line $L( x) =x$ approximates $f( x) =\sin x$ .

TABLE 3

${L(x) =x}$	${f(x) =\sin x}$	Error: ${\vert x-\sin x\vert}$
$0.1$	$0.0998$	$0.0002$
$-0.3$	$-0.2955$	$0.0045$
$0.4$	$0.3894$	$0.0106$
$0.5$	$0.4794$	$0.0206$
$\dfrac{\pi }{4}\approx 0.7854$	$0.7071$	$0.0783$

(c) See Figure 11 for the graphs of $f( x) =\sin x$ and $L( x) =x$ .