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 )

b. graph ( f ) and ( l ) on the same set of axes. use the graphing tool to graph the functions.

c. based on the graphs in part (b), state whether linear approximations to ( f ) near ( a ) are
underestimates or overestimates

linear approximations to ( f(x)=3 - x^{2} ) near ( a=-1 ) are because the graph of ( l )
lies the graph of ( f ) near ( a ).

Explanation:

Step1: Analyze the concavity of \(f(x)\)

The function \(f(x)=3 - x^{2}\), its second - derivative \(f''(x)=-2<0\). A function with \(f''(x)<0\) is concave down.

Step2: Relate concavity to linear approximation

For a function \(y = f(x)\) that is concave down at \(x = a\), the tangent line (linear approximation \(L(x)\)) lies above the graph of the function near \(x=a\).

Answer:

Linear approximations to \(f(x)=3 - x^{2}\) near \(a=-1\) are overestimates because the graph of \(L\) lies above the graph of \(f\) near \(a\).