16. -/1 points details my notes scalc9 2.9.031. use a linear approximation (or differentials) to estimate…

Explanation:

Step1: Define the function

Let $f(x)=x^{3}$.

Step2: Choose a value of $a$

Choose $a = 2$ since it is close to $1.999$ and easy to calculate $f(a)$.

Step3: Calculate $f(a)$ and $f'(x)$

$f(a)=f(2)=2^{3}=8$. $f'(x)=3x^{2}$, so $f'(2)=3\times2^{2}=12$.

Step4: Use the linear - approximation formula

The linear - approximation formula is $L(x)=f(a)+f'(a)(x - a)$. Here $x = 1.999$, $a = 2$. $L(1.999)=f(2)+f'(2)(1.999 - 2)$. $L(1.999)=8+12\times(1.999 - 2)$. $L(1.999)=8+12\times(-0.001)$. $L(1.999)=8 - 0.012$.

Answer:

$7.988$