find the linearization l(x) at x = a. f(x)=2x^{3}-3x - 1 a = - 1 l(x)=□
find the linearization l(x) at x = a. f(x)=2x^{3}-3x - 1 a = - 1 l(x)=□
Answer
Explanation:
Step1: Find f(a)
Substitute (x = a=- 1) into (f(x)=2x^{3}-3x - 1). [ \begin{align*} f(-1)&=2(-1)^{3}-3(-1)-1\ &=2\times(-1)+3 - 1\ &=-2 + 3-1\ &=0 \end{align*} ]
Step2: Find f'(x)
Differentiate (f(x)=2x^{3}-3x - 1) using the power - rule ((x^n)^\prime=nx^{n - 1}). (f^\prime(x)=6x^{2}-3)
Step3: Find f'(a)
Substitute (x = a=-1) into (f^\prime(x)). [ \begin{align*} f^\prime(-1)&=6(-1)^{2}-3\ &=6 - 3\ &=3 \end{align*} ]
Step4: Use the linearization formula
The linearization formula is (L(x)=f(a)+f^\prime(a)(x - a)). Substitute (f(-1) = 0), (f^\prime(-1)=3) and (a=-1) into the formula. [ \begin{align*} L(x)&=0+3(x-(-1))\ &=3(x + 1)\ &=3x+3 \end{align*} ]
Answer:
(3x + 3)