Question

let ( f(x)=3 - x^{2} ) and let ( a = - 1 ). complete parts (a) through (d) below.
a. find the linear approximation ( l ) for the function ( f ) at the point ( a ).
( l(x)=2x + 4 )

Explanation:

Step1: Find the value of \(f(a)\)

Given \(f(x)=3 - x^{2}\) and \(a=-1\), substitute \(x = a=-1\) into \(f(x)\):
\(f(-1)=3-(-1)^{2}=3 - 1=2\)

Step2: Find the derivative of \(f(x)\)

Differentiate \(f(x)=3 - x^{2}\) with respect to \(x\). Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^\prime(x)=-2x\)

Step3: Find the value of \(f^\prime(a)\)

Substitute \(x = a=-1\) into \(f^\prime(x)\):
\(f^\prime(-1)=-2\times(-1)=2\)

Step4: Use the linear - approximation formula

The linear - approximation formula is \(L(x)=f(a)+f^\prime(a)(x - a)\). Substitute \(f(a) = 2\), \(f^\prime(a)=2\), and \(a=-1\) into the formula:

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Answer:

\(L(x)=2x + 4\)