4 is the cube root of 64. Use the Linear Approximation for the cube root function at a=64 with Δx=1.4 to estimate how much larger the cube root of 65.4 is, i.e., 65.4−−−−√3−64−−√3≈

Answer 1

Linear approximation f(x) = f(a) + f'(a) (x-a)

Here 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 = 65.4

f(65.4) = 4 + 1/3 (64)^-2/3 (65.4 - 64)

f(65.4) - 4 = 1/3 1.4 (64)^-2/3 = 1/3 1.4 1/16 = 1.4 / 48 = 0.0292

if you find the exact difference it is 0.028956