Let h be the function given by h(x) =x+x-2
²-1
We will investigate the behavior of
both the numerator and denominator of h(x) near the point where x = 1. Let
f(x)= x³ + x -2 and g(x)=x²-1. Find the local linearizations of f and g at a = 1,
and call these functions Lf(x) and Lg(x), respectively.
Lf(x) =
L₂(x) =
Explain why h(x) ≈
Lf(x)
Lg(x)
for a near a = 1.

Final answer:

The local linearizations of f(x) and g(x) at a = 1 are Lf(x) = 4x - 5 and Lg(x) = 2x - 2 respectively. The function h(x) ≈ Lf(x)/Lg(x) because the local linearizations provide a good approximation of the numerator and denominator of h(x) near x = 1.

Explanation:

The local linearization of a function at a given point is an approximation of the function using a linear equation. To find the local linearization of a function f at a = 1, we need to find the slope of the tangent line at a = 1, which is equivalent to finding the derivative of f at x = 1. By taking the derivative of f(x) = x³ + x - 2, we get f'(x) = 3x² + 1. Evaluating f'(1), we find that the slope of the tangent line at a = 1 is 4. Therefore, the local linearization of f at a = 1, denoted as Lf(x), is given by Lf(x) = f(a) + f'(a)(x - a), which becomes Lf(x) = -1 + 4(x - 1) = 4x - 5.

Similarly, to find the local linearization of g(x) = x² - 1 at a = 1, we need to find the slope of the tangent line at a = 1. The derivative of g(x) is g'(x) = 2x. Evaluating g'(1), we find that the slope of the tangent line at a = 1 is 2. Therefore, the local linearization of g at a = 1, denoted as Lg(x), is given by Lg(x) = g(a) + g'(a)(x - a), which becomes Lg(x) = 0 + 2(x - 1) = 2x - 2.

When investigating the behavior of the function h(x) = (f(x))/(g(x)) near the point x = 1, we can approximate h(x) using the local linearizations of f and g at a = 1. Near the point a = 1, h(x) ≈ Lf(x)/Lg(x) because Lf(x) and Lg(x) provide a good approximation of the numerator and denominator, respectively, of h(x). This approximation holds as long as x is close to 1.

Learn more about local linearizations of functions here:

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