Short and Sweet Calculus

3.12 Linear Approximations

Because \(\lim_{\Delta x\to0}\frac{f(a+\Delta x)-f(a)}{\Delta x}=f'(a)\), when \(\Delta x=x-a\) is close to zero, we have \[\frac{f(a+\Delta x)-f(a)}{\Delta x}\approx f'(a).\] Multiplying both sides by \(\Delta x\), we get \[f(\underbrace{a+\Delta x}_{x})-f(a)\approx f'(a)\underbrace{(x-a)}_{\Delta x}\tag{\small\ensuremath{\Delta x=x-a}}\] or \[\boxed{f(x)\approx f(a)+f'(a)(x-a).}\] This is called the linear approximation of \(f\) at \(x=a\). It is also called the tangent line approximation, because the right-hand side \(y=f(a)+f'(a)(x-a)\) is the equation of the tangent line to \(y=f(x)\) at \(\left(a,f(a)\right).\)

  • The function \(L(x)=f(a)+f'(a)(x-a)\) is called the linearization of \(f\) at \(x=a\).

Example 3.24. Use the linear approximaiton to estimate \(\sin(29^{\circ})\).

Solution

Because we know the exact value of \(\sin(30^{\circ})=\sin(\pi/6)\), we can use the linearization of \(f(x)=\sin x\) at \(x=\pi/6\) to approximate \(\sin(29^{\circ})\). The derivative of \(f\) is \[f(x)=\sin x\Rightarrow f'(x)=\cos x\] and so we have \(f(\pi/6)=1/2\) and \(f'(\pi/6)=\cos(\pi/6)=\sqrt{3}/2\).

The linearization of \(f\) at \(x=\pi/6\) is \[L(x)=\underbrace{\frac{1}{2}}_{f(\pi/6)}+\underbrace{\frac{\sqrt{3}}{2}}_{f'(\pi/6)}\underbrace{\Delta x}_{\left(x-\frac{\pi}{6}\right)}.\] Notice that \(\frac{d}{dx}\sin x=\cos x\) works only when \(x\) is in radian measure. So \(\Delta x\) is also in radians. Because \(1^{\circ}=\frac{\pi}{180}\text{ radian,}\) we have \[\begin{aligned} \sin(29^{\circ}) & \approx L\left(\frac{\pi}{6}-\frac{\pi}{180}\right)=\frac{1}{2}+\frac{\sqrt{3}}{2}\left(-\frac{\pi}{180}\right)\\ & \approx0.4849.\end{aligned}\] The ture value of \(\sin(29^{\circ})\) to 4 digits is 0.4848, respectively.


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