Question

find the linear approximation to the following function at the given point a.
f(x) = -4x² + x; a = 3
l(x) = □

Explanation:

Step1: Find \( f(a) \)

Substitute \( x = a=3 \) into \( f(x)=-4x^{2}+x \).
\( f(3)=-4\times3^{2}+3=-4\times9 + 3=-36 + 3=-33 \)

Step2: Find \( f^{\prime}(x) \) and \( f^{\prime}(a) \)

Differentiate \( f(x)=-4x^{2}+x \) using the power rule \( (x^{n})^\prime=nx^{n - 1} \).
\( f^{\prime}(x)=-8x + 1 \)
Substitute \( x = 3 \) into \( f^{\prime}(x) \):
\( f^{\prime}(3)=-8\times3+1=-24 + 1=-23 \)

Step3: Use the linear - approximation formula \( L(x)=f(a)+f^{\prime}(a)(x - a) \)

Here \( a = 3 \), \( f(a)=-33 \), \( f^{\prime}(a)=-23 \)
\( L(x)=-33-23(x - 3) \)
Expand the expression:
\( L(x)=-33-23x+69 \)
\( L(x)=-23x + 36 \)

Answer:

\( L(x)=-23x + 36 \)