a. write the equation of the line that represents the linear approximation to the following function at the…

a. write the equation of the line that represents the linear approximation to the following function at the given point a. b. use the linear approximation to estimate the given quantity. c. compute the percent error in the approximation, 100·|approximation - exact|/|exact|, where the exact value is given by a calculator. f(x)=2 - 2x² at a = 3; f(2.9) a. l(x)=

a. write the equation of the line that represents the linear approximation to the following function at the given point a. b. use the linear approximation to estimate the given quantity. c. compute the percent error in the approximation, 100·|approximation - exact|/|exact|, where the exact value is given by a calculator. f(x)=2 - 2x² at a = 3; f(2.9) a. l(x)=

Answer

Explanation:

Step1: Find the derivative of the function

The function is $f(x)=2 - 2x^{2}$. Using the power - rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=-4x$.

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

Given $a = 3$, then $f(3)=2-2\times3^{2}=2 - 18=-16$ and $f'(3)=-4\times3=-12$.

Step3: Write the linear approximation formula

The linear approximation formula is $L(x)=f(a)+f'(a)(x - a)$. Substituting $a = 3$, $f(3)=-16$ and $f'(3)=-12$ into the formula, we get $L(x)=-16-12(x - 3)$.

Step4: Simplify the linear - approximation formula

$L(x)=-16-12x + 36=-12x+20$.

Answer:

$L(x)=-12x + 20$