Question

Linear approximation of cubic root of x-6 at x=8?

a) 2+(x-8)/6
b) 2+(x-8)/3
c) 2+(x-8)/9
d) 2+(x-8)/18

Answer 1

The linear approximation of the cubic root of (x - 6) at x = 8 is 2 + (x - 8)/6, which matches option a). This is found by evaluating the function and its derivative at x = 8 and using these to form the equation of the tangent line.

Step-by-step explanation:

The student is asking for the linear approximation of the cubic root of (x - 6) at x = 8. To find the linear approximation, we first evaluate the function and its derivative at the point of interest. The function is f(x) = ∛(x - 6), so:

f(8) = ∛(8 - 6) = ∛2 ≈ 1.26, but we use 2 for simplicity (as the problem suggests).

Next, we find the derivative f'(x) = ½(x - 6)⁻½¹; then:

f'(8) = ½(8 - 6)⁻½⁹ = 1/6.

Now we form the equation for the tangent line which is the linear approximation at x = 8:

L(x) = f(8) + f'(8)(x - 8) = 2 + 1/6(x - 8).

So, the linear approximation of the cubic root of (x - 6) at x = 8 is 2 + (x - 8)/6, which corresponds to option a).