Question

Find the linear approximation to f(x)=10-3x^2 at a=-2

Answer 1

Solution:

Given the function;

$f(x)=10-3x^2$

The linear approximation formula is;

$y=f(x)=f(a)+f^{\prime}(a)(x-a)$

Where;

$a=-2$

Then, the derivative is;

$f^{\prime}(x)=-6x$$\begin{gathered} f(a)=f(-2)=10-3(-2)^2 \\ f(a)=10-3(4)_{} \\ f(a)=10-12 \\ f(a)=-2 \end{gathered}$

Also,

$\begin{gathered} f^{\prime}(a)=f^{\prime}(-2)=-6(-2) \\ f^{\prime}(a)=12 \end{gathered}$

Thus, the linear approximation is;

$\begin{gathered} y=-2+12(x-(-2)) \\ y=-2+12(x+2) \\ y=-2+12x+24 \\ y=f(x)=12x+22 \end{gathered}$

FINAL ANSWER:

$f(x)=12x+22$