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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.)

Respuesta :

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)

facundo3141592

We want to find the differential of a function and also approximate the rate of change for a given interval.

We will get:

[tex]\frac{df}{dx} = -18*cos(x)*sin(x)[/tex]

And the average rate of change in the interval [π/6, π/6 +0.1] is:

r = -8.191

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we have:

[tex]f(x) = 9*cos^2(x)[/tex]

Now we need to differentiate it, we will use the rule:

[tex]f(x) = g(h(x))\\f'(x) = g'(h(x))*h'(x)[/tex]

Where:

[tex]g(x) = 9*x^2\\h(x) = cos(x)[/tex]

Then we will have:

[tex]\frac{df}{dx} = 2*(9*cos(x))*(-sin(x)) = -18*cos(x)*sin(x)[/tex]

To estimate the rate of change between two values x = a and x = b, we must compute:

[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]

Here we will have:

a = π/6

b = π/ + 0.1

Replacing these we get:

[tex]r = \frac{9*cos^2(\pi /6 + 0.1) - 9*cos^2(\pi /6)}{\pi /6 +0.1 -\pi /6} = -8.191[/tex]

If you want to learn more, you can read:

https://brainly.com/question/24062595

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