Linear approximation?

Question

use an appropriate local linear approximation to estimate the value of the given quantity.

(3.02)^4

Answer 1

Working with

y = x^4

dy = 4x^3 dx

for a local linearization, we can assume

∆y = 4x^3 ∆x

for points near x=3

∆y = 4(27) ∆x = 108 ∆x

∆x = .02

∆y = (108)(0.02) = 2.16

So

(3.02)^4 ≈ 3^4 + ∆y = 81 + 2.16 = 83.16

Answer 2

The linear approximation is

L(x) = f(x0) + f'(x0)(x-x0)

where f(x) = x^4, x0=3, and x=3.02. f'(x) = 4x^3, f(3) = 81, and f'(3) = 4(3^3) = 108.

L(3.02) = 81 + 108(3.02 - 3) = 81 + 108*0.02 = 81 + 2.16 = 83.16.