Goal: Understand the idea of the linear approximation of a function \(f(x)\) at
\(x = a\)
\[f(x) \approx L(x) = f(a) + f'(a)(x - a),\]
and apply it to approximate the value of a function at a given point.
Contents
1. Linear Approximation
Example Problem 10.1: Linear Approximation of \(\sqrt{x}\)
Find the linear approximation of \(f(x) = \sqrt{x}\) at \(x = a = 1\) and use it to approximate \(\sqrt{1.1}\).
Solution: Recall the formula for the linear approximation:
\[ L(x) = f(a) + f'(a)(x - a) \]
Compute \(f(a)\) and \(f'(a)\):
\[ f(1) = 1, \quad f'(x) = \frac{1}{2\sqrt{x}} \implies f'(1) = \frac{1}{2} \]
Linear approximation formula:
\[ L(x) = 1 + \frac{1}{2}(x-1) \]
Approximation of \(\sqrt{1.1}\):
\[ \sqrt{1.1} \approx L(1.1) = 1 + \frac{1}{2}(0.1) = 1.05 \]
Example Problem 10.2: Linear Approximation of \(e^x\)
Find the linear approximation of \(f(x) = e^x\) at \(x = a = 0\) and use it to approximate \(e^{-0.05}\).
\( f'(x) = e^x, \quad f(0) = 1, \quad f'(0) = 1 \)
Linear approximation formula:
\[ L(x) = 1 + 1 \cdot (x-0) = 1 + x \]
Approximation of \(e^{-0.05}\):
\[ e^{-0.05} \approx L(-0.05) = 1 - 0.05 = 0.95 \]
Example Problem 10.3: Linear Approximation of \(\sqrt[3]{x}\)
Estimate \(\sqrt[3]{8.012}\) using the linear approximation at \(x = 8\).
\( f(x) = x^{1/3}, \quad f'(x) = \frac{1}{3}x^{-2/3}, \quad f(8) = 2, \quad f'(8) = \frac{1}{12} \)
Linear approximation formula:
\[ L(x) = f(8) + f'(8)(x-8) = 2 + \frac{1}{12}(x-8) \]
Approximation of \(\sqrt[3]{8.012}\):
\[ \sqrt[3]{8.012} \approx 2 + \frac{1}{12}(0.012) = 2.001 \]
Example Problem 10.4: Linear Approximation of \(\sin x\)
Estimate \(\sin(46^\circ)\) using the linear approximation at \(a = 45^\circ = \pi/4\).
Convert to radians: \(46^\circ = \pi/4 + \pi/180 \approx 0.8029 \text{ rad}\)
\( f(x) = \sin x, \quad f'(x) = \cos x, \quad f(\pi/4) = \frac{\sqrt{2}}{2}, \quad f'(\pi/4) = \frac{\sqrt{2}}{2} \)
Linear approximation formula:
\[ L(x) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \left(x - \frac{\pi}{4}\right) \]
Approximation of \(\sin(46^\circ)\):
\[ \sin(46^\circ) \approx L(\pi/4 + \pi/180) = \frac{\sqrt{2}}{2} + \frac{\pi\sqrt{2}}{360} \]
2. Differentials
Let \(f\) be differentiable on an interval containing \(x\).
A small change in \(x\) is denoted by the differential \(dx\).
The corresponding change in \(f\) is approximated by the differential \(dy\).
\[
\Delta y = f(x + dx) - f(x) \approx dy = f'(x) \, dx
\]