Question

question 28
consider the function $f(x)=\sqrt3{x}$ near $x = 8$.
find the linear approximation error when using the linear approximation to estimate $\sqrt3{7.85}$.

• -0.00008
• 1.9875
• -1.9875
• 0.00008

Explanation:

Step1: Define function and point

$f(x)=\sqrt[3]{x}=x^{\frac{1}{3}}$, $a=8$, $x=7.85$

Step2: Compute $f(a)$ and $f'(a)$

$f(8)=\sqrt[3]{8}=2$
$f'(x)=\frac{1}{3}x^{-\frac{2}{3}}$, so $f'(8)=\frac{1}{3}(8)^{-\frac{2}{3}}=\frac{1}{3\times4}=\frac{1}{12}\approx0.083333$

Step3: Linear approximation formula

$L(x)=f(a)+f'(a)(x-a)$
$L(7.85)=2 + \frac{1}{12}(7.85-8)=2 + \frac{1}{12}(-0.15)=2-0.0125=1.9875$

Step4: Compute actual value

$f(7.85)=\sqrt[3]{7.85}\approx1.98758$

Step5: Calculate approximation error

$\text{Error}=f(x)-L(x)$
$\text{Error}\approx1.98758 - 1.9875=0.00008$

Answer:

0.00008