Finding the Linear Approximation to a Function

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

Solution (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 \(f( x) =\sin x\).

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