Final answer:
The linearization L(x) of the function at a given point is obtained by evaluating both the function and its derivative at that point, and then using the linearization formula to construct the linear approximation.
Step-by-step explanation:
The linearization L(x) of the function f(x) = x³ - x² + 5 at a = -3 is found by using the formula:
L(x) = f(a) + f'(a)(x - a)
We first calculate the value of the function and its derivative at a = -3.
- f(-3) = (-3)³ - (-3)² + 5 = -27 - 9 + 5 = -31
- The derivative of the function is f'(x) = 3x² - 2x, therefore:
- f'(-3) = 3(-3)² - 2(-3) = 27 + 6 = 33
Using these values, the linearization at a = -3 is:
L(x) = -31 + 33(x + 3)