find the linearization ( l(x) ) at ( x = a ).\n\n( f(x)=-2 x^{3}+5 x + 1 quad a = 2 )\n\n( l(x)=square )
find the linearization ( l(x) ) at ( x = a ).\n\n( f(x)=-2 x^{3}+5 x + 1 quad a = 2 )\n\n( l(x)=square )
Answer
Explanation:
Step1: Find the value of ( f(a) )
Substitute ( x = a = 2 ) into ( f(x)=-2x^{3}+5x + 1 ). [ \begin{align*} f(2)&=-2\times(2)^{3}+5\times2 + 1\ &=-2\times8 + 10+1\ &=-16 + 10+1\ &=-5 \end{align*} ]
Step2: Find the derivative ( f^{\prime}(x) )
Differentiate ( f(x)=-2x^{3}+5x + 1 ) using the power rule ( (x^{n})^\prime=nx^{n - 1} ). ( f^{\prime}(x)=-2\times3x^{2}+5=-6x^{2}+5 )
Step3: Find the value of ( f^{\prime}(a) )
Substitute ( x = a = 2 ) into ( f^{\prime}(x) ). [ \begin{align*} f^{\prime}(2)&=-6\times(2)^{2}+5\ &=-6\times4+5\ &=-24 + 5\ &=-19 \end{align*} ]
Step4: Use the linearization formula ( L(x)=f(a)+f^{\prime}(a)(x - a) )
Substitute ( f(2)=-5 ), ( f^{\prime}(2)=-19 ), and ( a = 2 ) into the formula. [ \begin{align*} L(x)&=-5+( - 19)(x - 2)\ &=-5-19x + 38\ &=-19x+33 \end{align*} ]
Answer:
( L(x)=-19x + 33 )