Suppose that we don't have a formula for g(x) but we know that g(1) = −5 and g'(x) =sqrt. x2 + 8 for all x. Use a linear approximation to estimate g(0.9) and g(1.1).

Answer 1

g'(x) = (x² + 8)^1/2

so g'(1) = (1² + 8)^1/2 = (1 + 8)^1/2 = 9^1/2 = 3

The linearization of the function g around x = 1 is given by:

L(x) - g(1) = 3(x - 1)

⇒ L(x) -(-5) = 3x - 3

⇒ L(x) + 5 = 3x - 3

⇒ L(x) + 5 - 5 = 3x - 3 - 5

⇒ L(x) = 3x - 8

Thus, using this linear approximation, estimates for g(0.9) and g(1.1) are given by:

g(0.9) ≈ L(0.9) = 3 * 0.9 - 8 = 2.7 - 8 = - 5.3

g(1.1) ≈ L(1.1) = 3 * 1.1 - 8 = 3.3 - 8 = - 4.7