Question

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

Answer 1

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`