Answer :
The exact difference is 0.022916
Given that the cube root of 64 is 4
We need to find how much larger is the cube root of 65.1 using the linear approximation.
Linear approximations:
They are ways to define a line tangent to a curve. At a particular range near the tangent point, the line can serve as an approximation for the function. However, linear approximations are determined through a combination of differentiation.
Here, when the tangent point is defined at x=a
then the linear approximation of a function is defined as f(x) ≈ f(a)+f′(a)(x−a)
Consider a = 64 f(x) = x^(1/3)
f'(x) = 1/3 x ^(-2/3)
So linear approximation is f(x) = 4 + 1/3 (64)^(-2/3) * (x-64)
x = 65.1
f(65.1) = 4 + 1/3 (64)^(-2/3) * (65.1 - 64)
f(65.1) - 4 = 1/3 1.1 (64)^(-2/3) = 1/3 * 1.1 * 1/16 = 1.1 / 48 = 0.0229
The exact difference is 0.022916
Learn more about linear approximation here
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