Chapter 3: Applications of Differentiation

Section 3.4: Differentials and the Linear Approximation

Example 3.4.1

$$ a)  Approximate $\sqrt{17}$ by using the differential of the function $f\left(x\right)\=\sqrt{x}$.   b)  How accurate is this approximation? $$ Solution $$ Part (a): Apply the linear approximation: $f\left(x\+\mathrm{dx}\right)$ $\doteq f\left(x\right)\+\mathrm{df}$ $f\left(16\+1\right)$ $\doteq f\left(16\right)\+f\prime \left(16\right)\cdot 1$ $$ Define the function $f$  •  Control-drag (or type) $f\left(x\right)\=\sqrt{x}$  •  Context Panel: Assign Function $f\left(x\right)\=\sqrt{x}$$\stackrel{\text{assign as function}}{\to }$$f$ $$ Calculate $\sqrt{17}$ accurately •  Type $f\left(17\right)$ and press the Enter key. •  Context Panel: Approximate≻10$$ $$$f\left(17\right)$ $\sqrt{17}$ $\stackrel{\text{at 10 digits}}{\to }$ $4.123105626$ (1) Linear approximation to $\sqrt{17}$  •  Type $f\left(16\right)\+f\prime \left(16\right)\cdot 1$ and press the Enter key. •  Context Panel: Approximate≻10 $f\left(16\right)\+f\prime \left(16\right)\cdot 1$ $\frac{33}{32}\sqrt{16}$ $\stackrel{\text{at 10 digits}}{\to }$ $4.125000000$ (2) $$ Part (b): Test the accuracy of the linear approximation: $$ Absolute error •  Absolute error: $\left|f\left(17\right)-\left(f\left(16\right)\+f\prime \left(16\right)\cdot 1\right)\right|$ $\left|-\right|$ $0.001894374$ (3) Relative error •  Relative error: $\left|\frac{f\prime \left(16\right)\cdot 1}{f\left(17\right)}\right|$ $\frac{}{}$ $0.0004594531821$ (4) Percent error •  Percent error:  $\left|\frac{f\prime \left(16\right)\cdot 1}{f\left(17\right)}\right|\cdot 100$  $\cdot 100$ = $0.04594531821$$$ $$

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