If f (x) = 9 cos2 (x), compute its differential df. df = (-18cos (x) sin (x)) dx correct: your answer is correct. approximate the change in f when x changes from x = π 6 to x = π 6 + 0.1. (round your answer to three decimal places.) δf =.738 incorrect: your answer is incorrect. approximate the relative change in f as x undergoes this change. (round your answer to three decimal places.)

Given: f (x) = 9 cos (2x)

The differential is

df = - 18 sin (2x) dx

When x varies from π/6 to π/6 + 01, then dx = 0.1.

The change in f is

δf = - 18 sin (π/3) * (0.1) = - 1.5588 ≈ - 1.559

If we compute the change in f directly, we obtain

f (π/6) = 9 cos (π/3) = 4.5

f (π/6 + 0.1) = 9 cos (π/3 + 0.2) = 2.6818

δf = 2.6818 - 4.5 = - 1.6382 ≈ - 1.638

Direct computation of δf is close to the actual value but in error.

The two results will be closer as dx gets smaller.

Answer:

δf = - 1.559 (correct answer)

δf = - 1.638 (approximate answer)