Question

Use a linear approximation to estimate the number (8.07)^2/3.

Answer 1

So let f(x) = x^(2/3)
Then let f'(x) = 2/3 x^(-1/3) = 2 / (3x^(1/3))

When x = 8,
f(8) = 8^(2/3) = 4
f'(8) = 2 / (3*8^(1/3)) = 1/3

So near x = 8, the linear approximation is
f(x) ≈ f(8) + f'(8) (x - 8)
f(x) ≈ 4 + 1/3 (x - 8)

So the linear approximation for x = 8.03 is...
f(8.03) ≈ 4 + 1/3 (8.03 - 8)
f(8.03) ≈ 4 + 1/3 (0.03)
f(8.03) ≈ 4.01

8.03^(2/3) ≈ 4.01

Answer 2

`8.07^(2/3)` ≈ 4.0233...

Explanation:

First we identify the function:
`f(x)=\sqrt[3]{x^(2) }`

then we take the firts derivative:
`f(x)=\frac{2}{3\sqrt[3]{x}}`

Then we take a starting point a=8, so the function has the value:

`f(8)=\sqrt[3]{8^(2) }=4`

and the first derivative has the value:

`f(8)=\frac{2}{3\sqrt[3]{8}}=(1)/(3)`

Then consider the folowing relation:

f(x) ≈ f(a) + f'(a) (Δx); where Δx = x-a = 8.07-8 = 0.07

Finally we replace the values and find:

`8.07^(2/3)` ≈ 4 +(
`(1)/(3)` * 0.07)

`8.07^(2/3)` ≈ 4.0233...