Suppose the linear approximation for a function f(x) at a = 3 is given by the tangent line y = −3x + 12

Question

If g(x) =[f(x)]², find the linear approximation for g(x) at a = 3.

L(x)=

Answer 1

Based on the linear approximation for the function f at a = 3, we know that:

f'(3) = -3 (slope of the tangent line)

f(3) = -3*3 + 12 = 3 (y-coordinate of the tangent line when x = 3)

To find the derivative of the function g, we must use the chain rule.

With that in mind, the derivative of the function g is given by:

g'(x) = f'(x) * 2f(x) = 2f(x)f'(x)

g(3) = [f(3)]² = 3² = 9

g'(3) = 2 * 3 * (-3) = -18

The linearization of the function g around a = 3 is given by:

L(x) - g(a) = g'(a)(x - a)

⇒ L(x) - g(3) = g'(3)(x - 3)

⇒ L(x) - 9 = -18(x - 3)

⇒ L(x) - 9 = -18x + 54

⇒ L(x) - 9 + 9 = -18x + 54 + 9

⇒ L(x) = -18x + 63